MODULATIONAL STABILITY OF MULTI-PULSES WITHIN THE

FUNCTIONALIZED CAHN-HILLIARD GRADIENT FLOW

By

Hayriye G¨u¸ckır C¸ akır

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Mathematics – Doctor of Philosophy

2019

ABSTRACT

MODULATIONAL STABILITY OF MULTI-PULSES WITHIN THE

FUNCTIONALIZED CAHN-HILLIARD GRADIENT FLOW

By

Hayriye G¨u¸ckır C¸ akır

The Functionalized Cahn-Hilliard (FCH) energy is a model describing the interfacial

energy in a phase separated mixture of amphiphilic molecules and a solvent. On a bounded

domain in R, the Euler-Lagrange equation for the mass constrained Functionalized Cahn-

Hilliard(FCH) free energy with zero functionalization terms is derived and a large family of

multi-pulse critical points is constructed. We show that the FCH energy with no functiona-

lization terms subject to a mass constraint has global minimizers over a variety of admissible

sets. We introduce a multi-pulse ansatz as the extensions of the periodic multi-pulse critical

points to R and establish the H2-coercivity of the second variation of the energy about

multi-pulse ansatz. Modulational stability and the dynamic evolution of the multi-pulse

ansatz with respect to the Π0-gradient flow are also addressed.

ACKNOWLEDGMENTS

I would like to express my deepest appreciation and gratitude to my advisor Dr. Keith

Promislow for his continuous personal and professional support throughout my years at

MSU. I would like to thank him for his contributions to my knowledge on partial differential

equations and applied mathematics with several courses he taught and useful discussions

during our regular meetings in spite of his busy schedule. The completion of this dissertation

would not be possible without his guidance, extensive knowledge and endless patience.

I would also like to thank my committee members Dr. Jeffrey Schenker, Dr. Russell

Schwab and Dr. Zhengfang Zhou for their helpful feedback and valuable time.

No words would be enough to express my gratitude to my parents and my sisters, Hale

and Esra, for their belief in me and unconditional support to pursue my dreams. I am deeply

indebted to my husband, Firat. Thank you for sharing my dreams and walking this journey

with me with joy for the last 14 years. Last but not least, I am grateful to my children, Miran

and Nehir, who have given me happiness and power keeping me motivated to complete this

work.

iii

TABLE OF CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Functionalized Cahn-Hilliard Free Energy . . . . . . . . . . . . . . . . . . . .
1.2 Description of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Main Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

1
1
4
8

Chapter 2 The Euler-Lagrange Equation . . . . . . . . . . . . . . . . . . . .
2.1 Derivation of the Euler-Lagrange Equation over Various Admissible Sets
. .
2.2 The Euler-Lagrange Equation with a small mass constraint . . . . . . . . . .
2.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4
2.2.5 Energy values at the critical points . . . . . . . . . . . . . . . . . . .

11
12
24
24
27
Solutions to the Euler-Lagrange Equation with a small mass constraint 29
Some Remarks
32
33

Chapter 3 Existence Of the Minimizers . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1 Existence of the Minimizers

Introduction: n-pulses

Chapter 4 Modulational Stability of n-pulses . . . . . . . . . . . . . . . . . .
4.1 H2-coercivity of the second variation of I . . . . . . . . . . . . . . . . . . . .
4.1.1
. . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 H2-coercivity of the second variation of I about n-pulse ansatz . . . .
4.1.3 H2-coercivity of L2
n . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Modulational Stability of n-Pulses . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Pulse Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37
37

46
46
47
49
59
65
77
87

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

iv

LIST OF FIGURES

Figure 1.1: When the hydrocarbons in different shapes introduced into a solvent, the
solvent particles create a cavity to avoid the solute(left)[Wiebe et al., 2012].
The simulation depicts packing of amphiphilic molecule at interface between
external solvent molecules and internal solvent(right). . . . . . . . . . . .

Figure 1.2: Figure 1.2a is the phase plane for the ODE (1.2.6) which demonstrates
the orbit (solid line) homoclinic to b− attaining its maxima at UM and
the periodic solutions (dotted lines) that are the solutions to the boundary
value problem(1.2.5). Figure 1.2b is an example of an n = 4 pulse solution
to (1.2.5)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(cid:16) L2

(cid:17) ∈ W s((cid:126)b) where (cid:126)b = (b, 0, 0, 0) mimic the behavior of the

Figure 2.1: Figure 2.1a represents a phase plane for the dynamical system given in
(2.2.2) extended to R for λ
 = 0. Figure 2.1b is a phase plane for the
dynamical system when λ
 (cid:54)= 0. The boundary conditions (cid:126)u(0) ∈ W u((cid:126)b)
and (cid:126)u
whole line system. The distance between the fixed points of the dynamical
systems is |b − b−| = ε while |u(0) − b| = O(e−κ
L2
ε ) (cid:28) ε. However, for
the clarity of the graph both distances are depicted in similar lengths. . .

ε

2

7

30

36

Figure 2.2: The reduced energy, I(Φn), versus mass constraint M . The blue lines
demonstrate the energy values at the critical points, Φn, for n = 0, 1
2, 2.
The red lines represent the infimum of the energy values over all the blue
lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 4.1: Figure (a) is a depiction for the spectrum of Ln(p). In the descending
order λ0 > 0 = λ1 > λ2 > ... (red disks) are the eigenvalues of L1(p) =
L. Ln(p) has n associated eigenvalues(black crosses) to each localized
eigenvalue of L1(p) such that |λk−λk,j|j=1,...,n decays exponentially with
growing pulse separation. Figure (b) demonstrates the spectrum for L2

2 , 1, 3

n(p). 53

v

Chapter 1

Introduction

1.1 Functionalized Cahn-Hilliard Free Energy

Amphiphilic molecules are chemical compounds consisting of a hydrophobic group and a

hydrophilic group, such as lipids and surfactants. When a molecule with an amphiphilic

structure is introduced to a solvent(water), the hydrophobic group may alter the structure

of the solvent which causes an increase in the free energy of the system. As a respond to this

change the system minimizes the contact between the hydrophobic group with formation

of bilayer interfaces and pores. Network formations differ from single layer interfaces that

occur in binary metals and other purely hydrophobic blends. While single layer interfaces

separate two distinct phases from each other, bilayers separate one phase by thin sheets of

other phase.

The Cahn-Hilliard(CH) free energy has been used broadly to model single layer interfaces

in hydrophobic blends. In 1958, the CH free energy was proposed as

(cid:90)

Ω

E(u) =

|(∇u)|2 + W (u)dx,

ε2
2

(1.1.1)

in [Cahn and Hilliard, 1958] to describe the free energy of an interface occurred by a phase

separation in a binary mixture due to spinodal decomposition. Here on a fixed domain
Ω ⊂ Rn, u ∈ H1(Ω) is the volume fraction of one of the components in the binary mixture,

1

W : R → R is the free energy density of the mixture and ε is the thickness of the interface.

Figure 1.1: When the hydrocarbons in different shapes introduced into a solvent, the solvent
particles create a cavity to avoid the solute(left)[Wiebe et al., 2012]. The simulation depicts
packing of amphiphilic molecule at interface between external solvent molecules and internal
solvent(right).

Since CH was introduced, its minimizers, minimizers subject to a constraint and critical

points of the CH have been broadly studied. Although a double well with two unequal depth

minima is generic, W (u) is assumed to be a smooth double-well potential with two equal

depth minima at b±,i.e, W (b−) = W (b+) and a maxima at b− < b0 < b+ in most studies.

This assumption on the form of the potential does not affect the following minimization

(cid:27)

(u − b−)dx = c

(cid:90)

Ω

,

(1.1.2)

problem with a mass constraint

(cid:26)(cid:90)

min

Ω

|(∇u)|2 + W (u)dx : u ∈ H1(Ω),

ε2
2

2

Bdx.doi.org/10.1021/jp209088u|J.Phys.Chem.CXXXX,XXX,000–000TheJournalofPhysicalChemistryCARTICLE2.MODELThepartialmolarvolume,VX,ofasolute,X,inagivensolvent,S,isthederivativeoftheoverallvolume,V,withrespecttothenumberofmolesofsolute,nX.VX¼∂V∂nXP,Tð2ÞFordilutesolutions,thiscanbereplacedwithafinite-differenceexpressionVX¼VðnS,nXÞVðnS,0ÞnXð3ÞThatis,VXisdefinedbythedifferencebetweenthevolumeofsolutionandthevolumeofpuresolventdividedbynX.Weproposetouseconstant-pressuremoleculardynamics(MD)simulationstoobtainthisdifferenceforasinglesolutemoleculeX(nX=1/NA;NAistheAvogadroconstant)andanN-particlesolventN3S(nS=N/NA).ThevolumeofasingleparticleV(X)isthengivenbyVðXÞ¼VðXþN3SÞVðN3SÞð4ÞandVX=NA3V(X).Thismethodmimicstheexperimentalprocedureofdeterminingpartialmolarvolumesofstablecom-poundsandisequallyapplicabletocalculationofpartialmolarvolumesofshort-livedTSspecies,forwhichsuchexperimentalprocedureisnotfeasible.Figure1showsfourhydrocarbonsimmersedinamodelsolvent.Ineachcase,thesolventtrajectorydemonstratesaclearpatternofavoidanceduetotheshort-rangesolventsoluterepulsion,whichresultsintheformationofacavityaroundthesolute.Theincrementalincreaseinthevolumeoftheoverallsystemduetocavityformationisdescribedbyeq4.Thesizeandtheshapeofthecavitydependonthegeometryofthesolute,thestrengthandtypeofthesolutesolventinteractions,aswellastemperatureandpressure.Volumefluctuationsinaconstant-pressureMDrunarequitesignificantandexceedthevalueofanincrementalvolumeincreaseduetoasinglemolecule.(SeeFigure2.)However,theaveragevolumefluctuatesmuchless,andthestandarderrorofthecumulativeaveragedecreasesasthesquarerootofthelengthofthetrajectory.AscanbeseeninFigure3,theerrorscanbereducedtoanacceptablelevelifanMDrunissufficientlylong.3.COMPUTATIONALDETAILSMDcalculationswereperformedusingtheGROMACSpackage9forasystemof256solventmoleculesinacubicboxwithperiodicboundaryconditions.Thesystemwasmaintainedataconstantpressureof0.1MPaandaconstanttemperaturematchingexperimentalconditionsusingBerendsentemperatureandpressurecoupling.10TheMDtrajectorieswereobtainedusingleapfrogintegrationwith1fstimestepwithinteractioncutoffradiusFigure1.Hydrocarbonsimmersedinamodelsolvent(clockwisefromthetopleftcorner):hexane,cyclopentadiene,benzene,andtoluene.Thesolventtrajectoryisrepresentedbyanoverlayofsolventconfigurationsacquiredatdifferentinstantsoftime.Solventparticles(white)avoidthesolute,thusformingacavityofthematchingsizeandshape.Figure2.InstantaneousandaverageMDvolumesofasystemof256cyclopentadienemolecules.Theincrementalcontributionfromasinglemoleculeestimatedbytheexperimentalmolarvolumeof82cm3/molisca.0.14nm3.Largeamplitudefluctuationsoftheinstantaneousvolumearesomewhatstabilizedbyaveragingover10psintervals.Furtherimprovementisreachedbyusingacumulativeaverage.Chem.Soc.Rev.Thisjournalis©TheRoyalSocietyofChemistry2018atomisticforcefieldsaremostsuitableforsimulatingtheself-assemblyofbio-inspiredmolecules.Comparativestudiesaresparseinthiscontext.AnexhaustivesetofMDsimulationsonthefoldingoftetra-andpentapeptides,performedusingCHARMM,AMBER,aswellasOPLS,showedclearinconsistenciesbetweentheseforcefields.26Asimilarobservationwasmadeinastudyonthedimerizationpropensityofaminoacidsidechainresidues.27Packingmotifs.Keepingtheaboveconsiderationsinmind,AAsimulationshavecontributedalottowardourunderstandingofthepackingmotifsofsupramolecularassemblies.Typicalsystemsetupsrangefromsmall-scalestudiesonmonomer–monomerinteractions,tolargerscaleself-assemblysimulationsandthestudyofpre-formedaggregates.Atthesmallestscale,i.e.monomersanddimers,AAsimulationsaremostlyusedtoimproveorvalidatetheunderlyingforcefield.Forinstance,Bejagametal.developedanAAmodelfor1,3,5-benzenetricarboxamide(BTA)usingQMcalculationsofgasphasedimersasareference.28TheenergyandstructureoftheBTAdimerattheQMlevelcouldbecapturedbytheAAmodel.TheAAmodelwasthenusedtostudydimerizationinnonane,andadimerfreeenergyof13kcalmol1wasobtained.Attachmenttoapre-constructedfibrewasfoundtobemorefavourable,demonstratingcoopera-tivityoftheprocess.Anotherniceexampleofforcefielddevel-opmentandvalidationistheworkofSasselietal.29TheauthorsdescribeasystematicparameterizationoftheFmocmoietyusingCHARMM,basedonQMdata(monomerinsolvent,dimer,torsionalpotentials)aswellasexperimentaloctanol/waterparti-tioningcoefficients.Validationofthemodelcamefromself-assemblyofdifferenttypesofFmoc-peptidesinwater,oneformingfibres,theothersphericalaggregates,bothinagree-mentwithexperiment.Inadditiontoforcefielddevelopment,smallscalesimulationscanbeveryusefultodetectpossiblebindingandstackingmodesofthebuildingblocks.30However,thequestionalwaysremainswhetherthesemodesarerepresen-tativefortheself-assembledstate.Initialself-assemblypathways.Theessenceofsupramolecularsystemsisthattheyformviaself-assembly.TocapturethisprocessinAAsimulationsisnottrivial,giventhelimitsintimeandlengthscalesthatcanbeachieved.Thetimescalesexploredareusuallylimitedtothenanosecondrange,withsystemsizestypicallycontainingafew100monomers.MostoftheAAself-assemblysimulationsreportedtodatearethereforenotleadingtowell-definedstructures,butoftenresultintheformationofsmallirregularaggregates.Despitethiscaveat,self-assemblysimulationsareimportantforshowingpossiblestackingmotifsinsidetheaggregateandrevealingthedrivingforcesoftheinitialgrowthprocess.Quiteanumberofstudiesdealwiththeself-assemblyofshortpeptides31–35orpeptideconjugates.36–38Alreadysuchsimplesystemsgiverisetoarichbehaviourintermsoftheinitialself-assemblykinetics,exem-plifiedbytheformationofunexpectedalpha-helicalinter-mediatestructuresofbeta-fibrilformingpeptides,31andthestrongeffectofpeptiderigidityontheorderinsidetheaggre-gatesformed.33Tofurthercomplicatetheissue,thedrivingforcesappearhighlysystem-dependent.Thep–p-stackingwasfoundtobeimportantindrivingtheinitialself-assemblyofpeptide–drugconjugatesintofibres,36whereasfibreformationintheabsenceofaromaticresidueswasobservedforothersmallpeptides.32OneofthemostinterestingAAstudiesonself-assemblyhasbeenreportedbyGarzonietal.39TheystudiedthemechanismofgrowthofBTAfibrilsinaqueousconditions.Basedonsimulationsofself-assembledshortnon-equilibriumaggregatesofdifferentlength,estimatesofthemonomer‘freeenergy’instacksofdifferentlengthcouldbeobtained.Theresultsindicateacooperativemechanismforsupramolecularpolymergrowth,whereacriticalsizemustbereachedintheaggregatesbeforeemergenceandamplificationoforderintotheexperimentallyobservedfibers.DetailedanalysisofthesimulationdatasuggeststhatH-bondingisamajorsourceofthisstabilizationenergy.Theworkprovidesevidenceforthekeydrivingforceofhydrogenbondinginenhancingthepersistencyofmonomerstackingandamplifyingtheleveloforderintothegrowingsupramolecularpolymer.Studiesintomoleculardrivingforcescanevenhavepracticalimpactinpopularculturesuchasavant-gardecuisine,asthe‘spherification’ofliquidfoodonthenanoscalewasfoundtobedrivenbycalcium-inducedaggregationofalginate.40Aniceexampleofthedifficultythatsimulationsattheall-atomlevelfacetoobtainequilibratedself-assembledstructuresisprovidedbytheworkofHaverkortetal.41Theyperformedextensiveself-assemblysimulationsofamphi-pseudoisocyanine,adyethatformshighlyordered,single-walledcylindricalJ-aggregatesinexperiment,seeFig.3.Thesimulatedstructure,althoughinthecorrectmorphology,containedahighamountofdisorder,despitemultiplecyclesofsimulatedannealing.Note,however,thatthepreciselevelofdisorderisnotknown,asthisisdifficulttoassessexperimentally.Itisconceivablethattheidealizedstructuresoftenusedtoillustratethepackingofmoleculesinsidefibresareunrealisticandoverlyordered.Idealizedstartingstructures.Thelackofknowledgeaboutthepackingdetailsofthesupramolecularaggregatesisoneofthemainreasonstoresorttoall-atomsimulations.Theideaofthisso-called‘‘top-down’’simulationapproachisthat,startingfromFig.3Self-assemblyofamphiphiliccyaninedyesintotubestructure.Thedyemoleculeisamphi-pseudoisocyanine,acyaninedyewithtwohydro-phobictails(CH3andC18H37).Thedyemoleculesaredepictedwiththetailspurple;thenitrogenatomsaredarkblue;thearomaticcarbonsarepink;thelinkerbetweenthearomaticringsislightblue;andhydrogensconnectedtoaromaticcarbonsarewhite.Perchloratecounterionsareshowningreen.Forthewatermolecules,oxygenatomsareredandhydrogenatomsarewhite.Figureadaptedwithpermissionfromref.41.Copyright2013AmericanChemicalSociety.ReviewArticleChemSocRevOpen Access Article. Published on 24 April 2018. Downloaded on 25/04/2018 02:04:47.  This article is licensed under a Creative Commons Attribution 3.0 Unported Licence.View Article Onlinesince this problem is equivalent to minimizing

(cid:90)
(cid:90)

˜E(u) =

=

ε2
2
ε2
2

Ω

Ω

(cid:90)

Ω

(u − b−)dx

|(∇u)|2 + W (u)dx − λ

|(∇u)|2 + W (u)dx − cλ,

(1.1.3)

where λ is a Lagrange multiplier and so cλ is only a fixed quantity. The functional ˜E is the
same as E with the well W replaced by W (u) + λu. An appropriate choice of λ will render

W (u) + λu a double well with equal depth minima. The critical points of the CH are the
solutions to δE

δu = 0 where

δE
δu

= ε2∆u − W(cid:48)(u),

(1.1.4)

is the variational derivative with respect to L2 inner-product. Changing variables into the

inner variables z = z

ε , at the leading order the critical point equation becomes

z u − W(cid:48)(u) = λ.
∂2

(1.1.5)

From the phase-plane analysis, the critical points of the CH subject to a mass constraint are

heteroclinic orbits or single layer interfaces which are the co-dimension 1 interfaces between

two distinct phases. Further, the single layer interfaces are the minimizers of the CH subject
to a mass constraint. The Γ-convergence as ε → 0 for the single layers structures of CH free

energy to a scaled surface area was established in [Modica, 1987] and [Sternberg, 1988]. A

model for the network formation in amphiphilic mixtures motivated by small-angle X-ray

scattering (SAXS) data was introduced in [Gompper and Schick, 1990] and

[Teubner and Strey, 1987] adding a higher order term in the CH free energy. Based on

these models, The FCH free energy was developed in [Promislow and Wetton, 2009] and in

3

[Gavish et al., 2011] as a mathematical model to describe the interfacial energy in a phase-

separated mixture with amphiphilic molecules.

F (u) =

ε2∆u − W(cid:48)(u)

(cid:17)2 − εp

(cid:18) η1ε2

2

(cid:19)

|∇u|2 + η2W (u)

dx,

(1.1.6)

(cid:16)

(cid:90)

1
2

Ω

where ε (cid:28) 1 is the ratio of amphiphilic molecule length to domain size and η1 > 0,η2 ∈ R.

For p = 1, the FCH corresponds to the strong functionalization while for p = 2 it is a

model for the weak functionalization. It is generic to assume that W (u) is a double-well

with two unequal depth minima at b±. Further, due to the quadratic term in FCH it is not

possible to rewrite the energy in terms of an equal-depth double-well. We also assume that
α± := W(cid:48)(cid:48)(b±) > 0. As a consequence, the dominant term in the FCH energy is the square
of the first L2 variational derivative of CH energy. Indeed the FCH energy can be viewed

as measuring L2 distance to the critical points of the associated CH energy. When η1 = 0

and η2 = 0, all critical points of CH free energy are the global minimizers of the FCH free

energy since

= 0 asserts that F (u) = 0. In [Promislow and Zhang, 2013], the existence

δE
δu

of global minimizers was established over a variety of admissible function space for a general

class of high order energies such as FCH free energy. Further, the authors showed that the

the critical points of CH are the critical points of the higher order energies.

1.2 Description of the Problem

Let a small amount of a polymer(soap) be added to a solvent(water) in a container with

impermeable walls and allow the system reach an equilibrium. Suppose that the mass of

the soap, m, scales with ε with a relation m = εM where ε > 0 is a small parameter and

4

M = O(1). We adapt the FCH free energy introduced in the previous section to model the
free energy of this system. For fixed L2 > 0, independent of ε, and Ω = [0, L2] ⊂ R, the

FCH free energy describing the free energy of the soap-water mixture is

(cid:90) L2

0

(cid:16)

1
2

(cid:17)2

I(u) =

ε2∂2

xu − W(cid:48)(u)

dx,

(1.2.1)

subject to the mass constraint

(cid:90) L2

0

m :=

(u − b−)dx = εM,

(1.2.2)

where M ∈ [0, M∗] ⊂ R for fixed M∗ and u satisfying non-flux boundary conditions. Here
the density function u ∈ H2(Ω) map Ω into R+ and the potential W (u) is an unequal

double well with two minima at b± for which W (b−) = 0 > W (b+) and a maxima at bM

where b− < bM < b+. For simplicity, we prefer converting our problem from macroscopic to

misroscopic level by converting spacial variables into inner variables. Introducing the inner

variable z = x

ε in (1.2.1), the inner scaling of the FCH free energy takes the form

(cid:90) L2

ε

(cid:16)

(cid:17)2
z u − W(cid:48)(u)
∂2

1
2

dz,

(1.2.3)

(u − b−)dz = M.

(1.2.4)

I (u) =

subject to the mass constraint

0

(cid:90) L2

ε

0

In this thesis, we aim constructing a special class of critical points of the inner scaling of the

FCH free energy subject to the mass constraint (1.2.4) as the possible minimizers of the free

5

energy. Apparently, I(u) ≥ 0 for all u ∈ H2(Ω). When I(u) is free of any constraints, the

solutions to

 ∂2

z u = W(cid:48)(u),

∂zu(0) = 0, ∂zu(

L2
ε ) = 0,

(1.2.5)

are global minimizers of I since they return I(u) = 0. To establish the existence of these

solutions we write the (1.2.5) as a non-linear system of first order differential equations

v(cid:48)
1 = v2
v(cid:48)
2 = W(cid:48)(v1)

(1.2.6)

and analyze the associated phase-plane diagram. The dynamical system (1.2.6) has two

saddle points at (b±, 0) and a center at the equilibrium point (bM , 0).(See Figure 1.2.) A

homoclinic solution is an orbit connecting a saddle point to itself. From the first integral of

(1.2.5)

1
2

(∂zu)2 = W (u) + Cn,

(1.2.7)

for any constant Cn. For the choice Cn = 0, we deduce that we have an orbit starting

at b−, hitting the uz = 0 axis at UM for which W (u) = 0 and joining back to b− by
reversibility. Thus, there exists a homoclinic solution, φh, converging to b− as z → ∞. The

homoclinic solutions do not satisfy the Neumann boundary conditions, having exponentially

small derivatives at z = 0 and z =

L2
ε . We can construct periodic solutions of (1.2.5) which

do satisfy the boundary conditions. The solutions satisfying the boundary conditions of

(1.2.5), φn, are the periodic solutions at the center starting at a point between b− and UM

on the axis uz = 0 and ending at a point on the same axis when z =

L2
ε making n = k+1
2 ,

6

k ∈ N tours. (See Figure 1.2.) By adjusting the value of the minimum of φn we adjust the

period, which we can tune to be an integral multiple of

so that the zero derivative points lie at z = 0 and z =

L2
ε . Translating the n-pulse periodic
L2
ε gives an exact solution of (1.2.5).

(a) Phase plane diagram

(b) An illustration for φ4

Figure 1.2: Figure 1.2a is the phase plane for the ODE (1.2.6) which demonstrates the orbit
(solid line) homoclinic to b− attaining its maxima at UM and the periodic solutions (dotted
lines) that are the solutions to the boundary value problem(1.2.5). Figure 1.2b is an example
of an n = 4 pulse solution to (1.2.5)

Upon this analysis, among many possible critical points of the mass constrained inner

scaling of the FCH free energy we desire to obtain a special class of those which are the

expansions of the minimizers of the unconstrained inner scaling of the FCH free energy,

n-periodic pulses.

Further, motivated by [Promislow, 2002] we survey the modulational stability and dynamical

evolution of the n-pulse structure of the inner scaling of the FCH energy. For this purpose,

we introduce the mass-preserving projection gradient flow of the FCH energy given in (1.2.3)

ut = −Π0

δI
δu

(u),

u(z, 0) = U0(z),

7

(1.2.8)

uzuUMb−UMb−φnzL2where the zero-mass projection, Π0, is given as Π0f := f −(cid:104)f(cid:105)Ω with (cid:104)f(cid:105)Ω := 1|Ω|

(cid:90)

Ω

f (x)dx

and observe that any critical point of the. As the zero-mass projection gradient of the FCH

energy evolves the mass of the initial data is preserved,

(cid:104)u(cid:105)Ω = 0,

d
dt

and the FCH energy given in (1.2.3), I, decreases,

I(u) ≤ 0.

d
dt

(1.2.9)

(1.2.10)

The mass-preserving gradient flow of the CH free energy modeling a phase separation process

in a binary mixture was analyzed in [Rubinstein and Sternberg, 1992].

1.3 Main Results

In Chapter 2, the main goal is to construct a class of the critical points of the inner scaling
of the FCH free energy subject to the mass constraint over the space of functions u ∈ H2(Ω)

satisfying the no-flux boundary conditions. In this purpose, we derive the Euler-Lagrange(E-

L) equation of the inner scaling of the FCH,



z − W(cid:48)(cid:48)(u)(cid:1)(cid:0)∂2
z u − W(cid:48)(u)(cid:1) = λε1,
(cid:0)∂2
(cid:17)
(cid:16) L2

= 0, ∂zu(0) = 0, ∂zu

∂3
z u(0) = 0, ∂3
z u

ε

(cid:17)

(cid:16) L2

ε

= 0,

(1.3.1)

8

over the admissible space

A1 := {u ∈ H2

0,

(cid:18)(cid:20)

(cid:21)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) L2

ε

0

L2
ε

(u − b−)dz = M, uz(0) = uz

(cid:18) L2

(cid:19)

ε

= 0}.

(1.3.2)

Motivating the corresponding scaling of the Langrange multiplier λε = ελ to the scaling of

the mass constraint, we use an asymptotic expansion to solve the E-L equation and obtain a

class of the critical points of the mass-constrained FCH energy I. Another result we present

in this chapter is that the mass constraint value has a significant impact on the minimizers

belonging to the class of the critical points we construct.

In Chapter 3, utilizing the classical tools from Calculus of Variations we obtain the

existence of the global minimizers of the FCH energy, I, subject to the mass-constraint
(1.2.4) over the admissible space A1.

In Chapter 4, we introduce n-pulse ansatz as the corrected extensions of the n-pulse

solutions of the E-L equation to the whole line R. We establish the H2-coercivity of the

second variation of the FCH energy I about n-pulse ansatz and further the H2-coercivity of

the second variation of the FCH energy I about periodic multi-pulses φn. With an application

of the result on modulational stability of the steady-state solutions of the gradient system

in [Promislow, 2002], we demonstrate that n-pulse ansatz, the steady-state solutions of the

mass-preserving projection gradient of I, is stable in the modulational sense. The evolution

equations of the pulse-locations and the background are derived as

¯λ(cid:48) = 0,
p(cid:48)
i = −α

3/2−

(cid:16)

e−√

α−(pi+1−pi) − e−√

α−(pi−pi−1)(cid:17)

+ O(δ2).

(1.3.3)

With an analysis of the evolution equations of the pulse locations we conclude that the

9

stationary solutions, equally spaced n-pulses(periodic) are spectrally stable. More significantly,

for a given initial data in an ε2-neighborhood of the n-pulse ansatz we recover the pulse
dynamics to O(δ2) where δ is exponentially small in ε. Moreover, the solutions remain
within a O(δ) neighborhood in H2 of the periodic n-pulses(equally spaced).

10

Chapter 2

The Euler-Lagrange Equation

In the calculus of variations, the Euler-Lagrange equations are used to construct the critical

points of a functional. In this section, we will derive the Euler-Lagrange equation for the

problem

(cid:90) L2

ε

(cid:16)

1
2

min
u∈A

(cid:17)2

(cid:90) L2

ε

z u − W(cid:48) (u)
∂2

dz

subject to

(u − b−)dz = M,

(2.0.1)

0

where the choice of the space of admissible functions A ⊂ H2(cid:16)(cid:104)

0

(cid:105)(cid:17)

0,

L2
ε

will be addressed

later. As described earlier, the problem is based on an experiment during which a polymer

is being added in to a solvent to form a dispersion in a fixed container with an impermeable

boundary and then the system is allowed to relax to reach its equilibrium. To model this,

we consider u ∈ H2(cid:16)(cid:104)

(cid:105)(cid:17)

0,

L2
ε

satisfying the mass constraint and the Neumann boundary

conditions due to impermeable membrane. With this description, we define the space of the

admissible functions

A1 := {u ∈ H2

(cid:18)(cid:20)

0,

L2
ε

(cid:21)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) L2

ε

0

(u − b−)dz = M, uz(0) = uz

(cid:18) L2

(cid:19)

ε

= 0}.

(2.0.2)

11

spaces

and

A2 =

L2
ε

ε

0

(cid:21)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) L2
(cid:21)(cid:19)(cid:12)(cid:12)(cid:90) L2

ε

0

(2.0.3)

(cid:27)

) = 0

.

(2.0.4)

L2
ε

Additionally, the free energy given (1.2.3) is well-defined on the following natural admissible

(cid:26)

A0 :=

u ∈ H2

(cid:18)(cid:20)

0,

(cid:27)

,

(u − b−)dz = M

(cid:26)

u ∈ H2

(cid:21)(cid:19)

(cid:18)(cid:20)

0,

L2
ε

(cid:18)(cid:20)

0,

L2
ε

∩ H1

0

(u − b−)dz = M, uz(0) = uz(

It may be easily observed that A2 ⊂ A1 ⊂ A0.

We will construct the Euler-Lagrange equation over all these spaces and discuss the

necessary boundary conditions. In the sequel, we will further see that the value of the mass

constraint has a considerable impact on the form of the actual minimizer(s).

2.1 Derivation of the Euler-Lagrange Equation over

Various Admissible Sets

The Euler-Lagrange equation characterizes the smooth critical points of the free energy
functional I(·). Recall that we consider u ∈ H2 satisfying the mass constraint

udz = M

ε

(cid:90) L2

and the boundary conditions uz(0) = uz

= 0 in the problem and our main purpose is

0

to construct the Euler-Lagrange equation for the critical points satisfying these conditions

in this section.

We first consider the largest space of admissible functions

(cid:26)

A0 :=

u ∈ H2

(cid:18)(cid:20)

0,

L2
ε

(cid:21)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) L2

ε

0

(cid:27)

.

(u − b−)dz = M

(2.1.1)

12

(cid:17)

(cid:16) L2

ε

Let u ∈ A0 be any critical point of the energy I(u) subject to the mass constraint and form
a curve u + τ v ∈ A0 for τ ∈ R and v ∈ A(cid:48)

0 where

(cid:26)

(cid:18)(cid:20)

A(cid:48)
0 =

v ∈ H2

0,

L2
ε

(cid:21)(cid:19)(cid:12)(cid:12)v ⊥ 1
(cid:27)
(cid:105)(cid:17)

0,

L2
ε

,

(2.1.2)

is the tangent plane to A0. The orthogonality condition is seen to be required by observing

that u + τ v ∈ A0 holds if and only if v ∈ H2(cid:16)(cid:104)

u + τ vdz = 0, this

and d
dτ

(cid:90) L2

ε

0

second requirement implies

0 =

d
dτ

=

(cid:90) L2
(cid:90) L2

0

ε

ε

(u + τ v) dz,

vdz,

(2.1.3)

and we deduce that v ⊥ 1.

0

We denote by i(τ ) the evaluation of I on the curve u + τ v

i(τ ) := I(u + τ v).

(2.1.4)

For the following calculations, let us assume that i(τ ) is differentiable at τ = 0 which will

be established in Theorem 2.1.2 later. Assuming that u is a critical point of I, i has zero
derivative at τ = 0, i.e. i(cid:48)(0) = 0. We formally calculate the variation of I

ε

(cid:90) L2
(cid:90) L2
(cid:90) L2

0

0

ε

ε

0

i(cid:48)(τ ) =

=

=

d
dτ

(cid:16)

(cid:17)2
z (u + τ v) − W(cid:48)(u + τ v)
∂2
(cid:17)(cid:16)
z (u + τ v) − W(cid:48)(u + τ v)
∂2
z v −(cid:16)
(cid:17)
z (u + τ v) − W(cid:48)(u + τ v)
∂2

∂2

1
2

(cid:16)
(cid:16)

dz,

z v − W(cid:48)(cid:48)(u + τ v)v
∂2

(cid:17)

dz,

z (u + τ v) − W(cid:48)(u + τ v)
∂2

(cid:17)

W(cid:48)(cid:48)(u + τ v)vdz.
(2.1.5)

13

Let τ = 0 and since i(cid:48)(0) = 0

ε

(cid:16)

(cid:90) L2

0 = i(cid:48)(0) =

(cid:17)
z u − W(cid:48)(u)
∂2
Further, if the critical point u ∈ H4(cid:16)(cid:104)

(cid:48).

0

for all v ∈ A0

∂2

z v −(cid:16)
(cid:105)(cid:17)

0,

L2
ε

(cid:17)
z u − W(cid:48)(u)
∂2

W(cid:48)(cid:48)(u)vdz,

(2.1.6)

we may twice integrate by parts the second

term in the integrand and obtain

0 := Gu(v) =

ε

0

(cid:90) L2
(cid:124)
(cid:16)
(cid:124)

−∂z

(cid:16)

(cid:16)

z u − W(cid:48)(u)
∂2

∂2
z

(cid:17)
z u − W(cid:48)(u)
∂2

v

(cid:17)
z u − W(cid:48)(u)
∂2
(cid:17)

z u − W(cid:48)(u)
∂2

(cid:17) −(cid:16)
(cid:123)(cid:122)
(cid:12)(cid:12)(cid:12) L2
(cid:16)
(cid:123)(cid:122)

+

A

0

ε

B

(cid:17)
W(cid:48)(cid:48)(u)

(cid:12)(cid:12)(cid:12) L2
(cid:125)

0

ε

,

∂zv

+

vdz

(cid:125)

(2.1.7)

for all v ∈ A(cid:48)
0. The map v (cid:55)→ Gu(v) is the weak formulation of the variational derivative
of the free energy I(u). The equality in (2.1.7) holds for all v ∈ A0 but different subspaces
A(cid:48)
0 afford information on different terms in the RHS of (2.1.7). In particular, if we choose v
from the subspace S(cid:48)

0 then the boundary terms B are 0

0 = {v ∈ C∞

0,

c

L2
ε

(cid:16)(cid:104)

(cid:105)(cid:17)(cid:12)(cid:12)(cid:12)v ⊥ 1} ⊂ A(cid:48)
(cid:17)(cid:16)
z − W(cid:48)(cid:48)(u)
∂2

(cid:17)
z (u) − W(cid:48)(u)
∂2
(cid:105)(cid:17)

we infer that

and we deduce that

(cid:90) L2

ε

0

for all v ∈ S(cid:48)
0.

By the density of S(cid:48)

(cid:16)
0 in L2(cid:16)(cid:104)

vdz = 0,

(2.1.8)

0 in A(cid:48)
(cid:16)

0,

L2
ε

(cid:17)(cid:16)
z − W(cid:48)(cid:48)(u)
∂2

(cid:17) ⊥ A(cid:48)

0,

z u − W(cid:48)(u)
∂2

14

(2.1.9)

(cid:26)

v ∈ H2(cid:16)(cid:104)

since A(cid:48)

0 =

0,

L2
ε

(cid:16)

(cid:27)
(cid:105)(cid:17)(cid:12)(cid:12)v ⊥ 1

we see that A(cid:48)⊥

0 = span{1} and hence
(cid:17)
z u − W(cid:48)(u)
∂2

= λε1,

(cid:17)(cid:16)
z − W(cid:48)(cid:48)(u)
∂2

(2.1.10)

0

(v(0), v(

for some Lagrange multiplier λε. Consequently, the relation (2.1.8) holds for all v ∈ A(cid:48)
even for those which are not in S(cid:48)
conclude that B = 0 to preserve the equality in (2.1.7). Since the trace map v ∈ A(cid:48)

0. This information implies that A = 0 in (2.1.7) and we
0 (cid:55)→
ε )) ∈ R4 is onto, we may choose v1, v2, v3, v4 for which this trace
L2
map yields the canonical basis e1, e2, e3, e4. These choices of v show that the critical point
u from the admissible space A0 satisfy the following boundary conditions,

L2
ε ), ∂zv(0), ∂zv(

z u − W(cid:48)(u)∂zu(cid:1)(cid:12)(cid:12)
z u − W(cid:48)(u)∂zu(cid:1)(cid:12)(cid:12)z=0= 0,
(cid:0)∂3
(cid:0)∂3
z u − W (u)(cid:1)(cid:12)(cid:12)
z u − W (u)(cid:1)(cid:12)(cid:12)z=0= 0,
(cid:0)∂2
(cid:0)∂2


Proposition 2.1.1. Any critical point of the problem (2.0.1) over A0 that lies in H4(cid:16)(cid:104)

We summarize the results obtained so far in the following proposition.

(2.1.11)

z=

L2
ε

z=

L2
ε

= 0,

= 0.

(cid:105)(cid:17)

0,

L2
ε

satisfies 

(cid:0)∂2
z − W(cid:48)(cid:48)(u)(cid:1)(cid:0)∂2
z u − W(cid:48)(u)(cid:1) = λε1,
z u − W(cid:48)(u)∂zu(cid:1)(cid:12)(cid:12)z=0 = 0,
z u − W(cid:48)(u)∂zu(cid:1)(cid:12)(cid:12)
(cid:0)∂3
(cid:0)∂3
z u − W (u)(cid:1)(cid:12)(cid:12)z=0= 0,
z u − W (u)(cid:1)(cid:12)(cid:12)
(cid:0)∂2
(cid:0)∂2

= 0.

z=

L2
ε

= 0,

z=

L2
ε

(2.1.12)

Another admissible space, which is actually our main focus of interest for the minimization

15

problem, is given by

(cid:26)

(cid:18)(cid:20)

(cid:21)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) L2

ε

0

L2
ε

A1 =

u ∈ H2

0,

(u − b−)dz = M, uz(0) = uz

(cid:18) L2

(cid:19)

ε

(cid:27)

= 0

,

(2.1.13)

where Neumann-boundary conditions uz(0) = uz

= 0 emulate no-flux boundary

(cid:17)

(cid:16) L2

ε

conditions due to the impermeable boundary. Following the same procedure we adopted
as deriving the Euler-Lagrange equation satisfied by any u ∈ A0, we achieve (2.1.8) for all
v ∈ A(cid:48)

1 where

(cid:26)

(cid:18)(cid:20)

A(cid:48)
1 =

v ∈ H2

0,

L2
ε

(cid:21)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)v ⊥ 1, vz(0) = vz

(cid:18) L2

(cid:19)

ε

(cid:27)

= 0

,

(2.1.14)

is the tangent plane to the admissible space A1.

Taking this into consideration and inserting the boundary conditions uz(0) = uz

(cid:17)

(cid:16) L2

ε

=

0 imposed in the admissible space A1 in (2.1.7) we obtain
(cid:17)
z (u) − W(cid:48)(u)
∂2

0 = ∂z

(cid:16)

v

(cid:12)(cid:12)(cid:12) L2

ε

0

,

(2.1.15)

A1 ∩ H4(cid:16)(cid:104)

(cid:105)(cid:17)
for all v ∈ A(cid:48)
1. Similar to the previous case, we conclude that any critical point u ∈
L2
ε

satisfies

0,



z − W(cid:48)(cid:48)(u)(cid:1)(cid:0)∂2
z u − W(cid:48)(u)(cid:1) = λε1,
(cid:0)∂2
(cid:17)
(cid:16) L2

= 0, ∂zu(0) = 0, ∂zu

∂3
z u(0) = 0, ∂3
z u

ε

(cid:17)

(cid:16) L2

ε

= 0.

(2.1.16)

16

The last natural admissible space we discuss in this section is

(cid:26)

(cid:18)(cid:20)

(cid:21)(cid:19)

(cid:18)(cid:20)

A2 =

u ∈ H2

0,

L2
ε

∩ H1

0

0,

L2
ε

(cid:21)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) L2

ε

0

(u − b−)dz = M, uz(0) = uz

(cid:18) L2

(cid:19)

ε

(cid:27)

= 0

,

(2.1.17)

which has the tangent plane

(cid:26)

(cid:18)(cid:20)

A(cid:48)
2 =

v ∈ H2

0,

L2
ε

(cid:21)(cid:19)

(cid:18)(cid:20)

0,

(cid:21)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)v ⊥ 1, vz(0) = vz

(cid:18) L2

(cid:19)

ε

L2
ε

∩ H1

0

(cid:27)

= 0

.

(2.1.18)

In a similar manner to the previous two cases, it can be easily demonstrated that any

minimizer u ∈ A2 ∩ H4([0,

L2
ε ]) solves the Euler-Lagrange equations



z u − W(cid:48)(u)(cid:1) = λε1,
z − W(cid:48)(cid:48)(u)(cid:1)(cid:0)∂2
(cid:0)∂2
(cid:16) L2
(cid:17)

= 0, ∂zu(0) = 0, ∂zu

u(0) = 0, u

ε

(cid:17)

(cid:16) L2

ε

= 0,

(2.1.19)

and no new boundary conditions arise. From now on, for simplicity of notation we let
A := A1.

Our earlier construction of the Euler-Lagrange equation assumed that i(τ ) is differentiable

at τ = 0. The following theorem provides a justification for this fact.

Theorem 2.1.2. Consider I(u) given (1.2.3). Then, for any u, v ∈ H2(cid:16)(cid:104)

(cid:105)(cid:17)

0,

L2
ε

i(τ ) = I(u + τ v),

(2.1.20)

is differentiable at τ = 0.

17

Proof. Take any u, v ∈ H2(cid:16)(cid:104)

(cid:105)(cid:17)

0,

L2
ε

and set

i(τ ) = I(u + τ v).

(2.1.21)

First, i(τ ) is finite for all τ since u ∈ H2(cid:16)(cid:104)

(cid:105)(cid:17)

0,

L2
ε

and the square of L2 functions lie in L1.

Let τ (cid:54)= 0. Evaluate the difference quotient for a.e. z

(cid:90) L2

ε

0

=

i(τ ) − i(0)

τ

(cid:0)∂2
z (u + τ v) − W(cid:48) (u + τ v)(cid:1)2

1
2

dz − 1

2

(cid:0)∂2
z u − W(cid:48) (u)(cid:1)2

τ

dz.

(2.1.22)

Inserting the Taylor series expansion of W(cid:48)(u + τ v) with integral remainder,

W(cid:48)(u + τ v) = W(cid:48)(u) + τ W(cid:48)(cid:48)(u)v +

W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds,

(2.1.23)

in to (2.1.22), and after some simplifications we find that the quotient reduces to

(cid:90) u+τ v

u

i(τ ) − i(0)

τ

=

1
τ

(cid:90) L2

ε

0

ε

0

1
τ

− 1
2

(cid:90) L2
−(cid:16)
(cid:90) L2

− τ

ε

=

=

(cid:90) u+τ v

(cid:19)2
W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds

1
2

(cid:18)
z v − W(cid:48)(u) − τ W(cid:48)(cid:48)(u)v −
(cid:16)
(cid:17)2
∂2
z u + τ ∂2
z u − W(cid:48) (u)
(cid:16)
(cid:17)
(cid:17)(cid:16)
∂2
z v − W(cid:48)(cid:48)(u)v
z u − W(cid:48)(u)
(cid:17)(cid:18)(cid:90) u+τ v
∂2
∂2
z u − W(cid:48)(u)
(cid:17)(cid:18)(cid:90) u+τ v
(cid:16)
∂2
z v − W(cid:48)(cid:48)(u)v
∂2

dz,

+

u

τ

u

u

(cid:18)(cid:90) u+τ v

W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds
(cid:19)
+ τ 2(cid:16)
z v − W(cid:48)(cid:48)(u)v
W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds
(cid:19)
∂2

u

W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds

dz+

+

+

(cid:19)2
(cid:17)2

dz,

(2.1.24)

Lτ (z)dz,

0

18

(cid:19)2

(cid:17)

(cid:18)(cid:90) u+τ v

u

+

W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds
(cid:19)
W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds

+ τ 2(cid:16)
(cid:19)(cid:21)
W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds

.

z v − W(cid:48)(cid:48)(u)v
∂2

+

(cid:17)2

+

(2.1.25)

. For the proof we first apply the triangle inequality to the

where we have introduced

(cid:34)

Lτ :=

1
τ

τ

(cid:17)(cid:16)
(cid:16)
z v − W(cid:48)(cid:48)(u)v
z u − W(cid:48)(u)
(cid:17)(cid:18)(cid:90) u+τ v
∂2
∂2
−(cid:16)
z u − W(cid:48)(u)
(cid:17)(cid:18)(cid:90) u+τ v
(cid:16)
∂2
z v − W(cid:48)(cid:48)(u)v
∂2
(cid:105)(cid:17)

−τ

u

u

L2
ε

We claim that Lτ ∈ L1(cid:16)(cid:104)
(cid:90) L2
(cid:90) L2

integrand and obtain

0,

ε

ε

|Lτ (z)|dz =

1
τ

0

0

+

+

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)τ
(cid:16)
(cid:17)(cid:16)
z u − W(cid:48)(u)
(cid:18)(cid:90) u+τ v
∂2
−(cid:16)
z u − W(cid:48)(u)
+ τ 2(cid:16)
∂2
(cid:16)

z v − W(cid:48)(cid:48)(u)v
∂2
z v − W(cid:48)(cid:48)(u)v
∂2

− τ

(cid:17)
(cid:19)2
W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds

z v − W(cid:48)(cid:48)(u)v
∂2
(cid:17)(cid:18)(cid:90) u+τ v
(cid:17)2
(cid:17)(cid:18)(cid:90) u+τ v

+

u

u

u

+

(cid:19)
W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds

W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds

(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)dz.

19

+

(cid:19)2(cid:12)(cid:12)(cid:12)(cid:12)
(cid:125)

+

W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds

1
τ

Taking

inside the integral and by the triangle inequality we have

0

0

u

1
τ

:=C

+

+

:=A

:=B

τ

u

(cid:90) L2

ε

(cid:90) L2

ε

|Lτ (z)|dz ≤

z v − W(cid:48)(cid:48)(u)v
∂2

(cid:123)(cid:122)
(cid:17)(cid:18)(cid:90) u+τ v
(cid:123)(cid:122)
(cid:17)2(cid:12)(cid:12)(cid:12)(cid:12)
(cid:125)
z v − W(cid:48)(cid:48)(u)v
∂2
(cid:17)(cid:18) 1

(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:16)
(cid:17)(cid:16)
(cid:124)
(cid:125)
(cid:123)(cid:122)
z u − W(cid:48)(u)
∂2
(cid:12)(cid:12)(cid:12)(cid:12)τ
(cid:18) 1
(cid:90) u+τ v
(cid:124)
W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds
(cid:12)(cid:12)(cid:12)(cid:12)(cid:16)
(cid:124)
z u − W(cid:48)(u)
∂2
(cid:12)(cid:12)(cid:12)(cid:12)τ
(cid:16)
(cid:124)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:16)
(cid:124)
(cid:105)(cid:17)
A, B, C, D and E are in L1(cid:16)(cid:104)
Adz ≤(cid:13)(cid:13)(cid:13)∂2

(cid:90) u+τ v
(cid:123)(cid:122)

z v − W(cid:48)(cid:48)(u)v
∂2

z u − W(cid:48) (u)

(cid:90) L2

ε

(cid:13)(cid:13)(cid:13)∂2

0,

L2
ε

(cid:123)(cid:122)

(cid:13)(cid:13)(cid:13)L2

τ

u

:=D

:=E

+

+

+

0

By the Lebesgue Dominated Convergence Theorem, it suffices to show that each of the terms

. For the first term, by the Holder’s inequality

W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds

dz.

+

(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:125)
(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:125)

(2.1.26)

(2.1.27)

(2.1.28)

(cid:13)(cid:13)(cid:13)L2

,

z v − W(cid:48)(cid:48) (u) v

,

(cid:17)
+(cid:13)(cid:13)W(cid:48)(cid:48)(u)v(cid:13)(cid:13)L2
(cid:17)
+(cid:13)(cid:13)W(cid:48)(cid:48)(u)(cid:13)(cid:13)L∞ (cid:107)v(cid:107)L2
(cid:13)(cid:13)(cid:13)W (n)(s)

.

Then, from the triangle inequality we have

(cid:90) L2

ε

0

Adz ≤(cid:16)(cid:13)(cid:13)(cid:13)∂2
≤(cid:16)(cid:13)(cid:13)(cid:13)∂2

z u

z u

(cid:13)(cid:13)(cid:13)L2
(cid:13)(cid:13)(cid:13)L2

+(cid:13)(cid:13)W(cid:48)(u)(cid:13)(cid:13)L2
+(cid:13)(cid:13)W(cid:48)(u)(cid:13)(cid:13)L2

(cid:17)(cid:16)(cid:13)(cid:13)(cid:13)∂2
(cid:17)(cid:16)(cid:13)(cid:13)(cid:13)∂2

z v

z v

(cid:13)(cid:13)(cid:13)L2
(cid:13)(cid:13)(cid:13)L2

(cid:13)(cid:13)(cid:13)L∞ ≤ α1 for
all s ∈(cid:2)−(cid:107)u(cid:107)L∞ − τ (cid:107)v(cid:107)L∞ ,(cid:107)u(cid:107)L∞ + τ (cid:107)v(cid:107)L∞(cid:3). Since u ∈ H2 implies (cid:107)u(cid:107)L∞ is bounded,

By the smoothness of W , for each n there exists an α1 > 0 such that

20

we conclude that A ∈ L1(cid:16)(cid:104)

(cid:105)(cid:17)

0,

L2
ε

that there exists a constant α1 such that

for each u, v ∈ H2(cid:16)(cid:104)
(cid:90) L2

Ddz ≤ α1τ.

ε

(cid:105)(cid:17)

0,

L2
ε

. The same arguments show

(2.1.29)

0

On the other hand, bounding B, C and E requires estimates on the integral remainder

term. From the smoothness of W and the compact range of u and u + τ v, there exists a
α2 > 0 such that |W(cid:48)(cid:48)(cid:48)(s)| ≤ α2 for s running over u to u + τ v. Then,

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1

τ

(cid:90) u+τ v

u

W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds

(cid:90) u+τ v

(cid:12)(cid:12)u + τ v − s(cid:12)(cid:12)ds.

(2.1.30)

By the change of variables t = u + τ v − s we obtain

(cid:90) u+τ v

u

(cid:12)(cid:12)u + τ v − s(cid:12)(cid:12)ds =

|t|dt ≤ τ 2
2

|v2|.

(2.1.31)

τ v

u

τ

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α2
(cid:90) 0

Inserting this in (2.1.30) we have

(cid:90) u+τ v

u

W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α2τ

2

(cid:12)(cid:12)v2(cid:12)(cid:12).

(2.1.32)

With these estimates we deduce that

Bdz ≤

W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds

(cid:13)(cid:13)(cid:13)(cid:13)2

,

L2

,

(2.1.33)

τ

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1
(cid:90) L2

ε

0

(cid:13)(cid:13)(cid:13)(cid:13)(cid:90) u+τ v
(cid:13)(cid:13)(cid:13)v2(cid:13)(cid:13)(cid:13)2

α2
2
2 (cid:107)v(cid:107)4
α2

u

L2

L4 .

≤ τ 4
4
τ 4
4

=

21

Here we controlled the L4 norm by H1 and L∞ norms using the estimate

(cid:90) L2

ε

0

(cid:107)v(cid:107)4

L4 =

v4dz ≤ (cid:107)u(cid:107)2

L∞ (cid:107)u(cid:107)2

L2 ≤ (cid:107)u(cid:107)2

H1 (cid:107)u(cid:107)2

L∞ ,

(2.1.34)

and further L∞ norm by H1 norm

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) L2

ε

z

(cid:12)(cid:12)(cid:12)(cid:12) ≤ 2(cid:107)v(cid:107)L2(cid:107)∂zv(cid:107)L2 ≤ (cid:107)v(cid:107)2

∂x(v2)dx

(cid:107)v(cid:107)2

L∞ = sup
z∈Ω

we obtain

L2 + (cid:107)∂zv(cid:107)2

L2 ≤ (cid:107)v(cid:107)2

H1 , (2.1.35)

(cid:107)v(cid:107)4

L4 ≤ (cid:107)v(cid:107)4

H1 ≤ C (cid:107)v(cid:107)4

H2 .

Utilizing this estimate in (2.1.33), we have

(cid:90) L2

ε

0

Bdz ≤ τ 4
4

2 (cid:107)v(cid:107)4
α2

H2 .

With the same arguments we obtain upper bounds for C and E such that

W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds

(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)dz,

(cid:90) L2

ε

(cid:90) L2

ε

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16)
(cid:90) L2

ε

z u − W(cid:48)(u)
∂2

u

τ

(cid:17)(cid:18) 1

(cid:90) u+τ v
(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)dz,
z u − W(cid:48) (u)
(cid:13)(cid:13)(cid:13)L2
+(cid:13)(cid:13)W(cid:48)(u)(cid:13)(cid:13)L2
∂2
(cid:13)(cid:13)(cid:13)H2
(cid:17)

+ α1

z u

,

z u

(cid:12)(cid:12)(cid:12)(cid:12)v2(cid:16)
(cid:16)(cid:13)(cid:13)(cid:13)∂2
(cid:16)(cid:13)(cid:13)(cid:13)∂2

0

(cid:107)v(cid:107)2
L4
(cid:107)v(cid:107)2
H2

(cid:17)

,

Cdz =

0

0

≤ τ α2
2
≤ τ α2
2
≤ τ α2
2

(2.1.36)

(2.1.37)

(2.1.38)

22

and(cid:90) L2

ε

0

(cid:90) L2

ε

0

Edz =

≤ τ 2α2
2

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)τ
(cid:16)
(cid:90) L2

ε

0

(cid:12)(cid:12)(cid:12)(cid:12)v2(cid:16)

z v − W(cid:48)(u)v
∂2

(cid:17)(cid:18) 1

τ

(cid:90) u+τ v
(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)dz,

u

z v − W(cid:48) (u) v
∂2

W(cid:48)(cid:48)(cid:48)(s) (u + τ v − s) ds

(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)dz,

(2.1.39)

≤ (1 + α1)

τ 2α2

(cid:107)v(cid:107)3

H2 .

Considering all these estimates for A, B, C, D and E and u, v ∈ H2(cid:16)(cid:104)

2

(cid:105)(cid:17)

(cid:17)(cid:16)(cid:13)(cid:13)∂2

0,

L2
ε

z v(cid:13)(cid:13)L2 +(cid:13)(cid:13)W(cid:48)(cid:48)(u)(cid:13)(cid:13)L∞ (cid:107)v(cid:107)L2

we conclude

(cid:17) ∈

(cid:16)(cid:13)(cid:13)∂2
z u(cid:13)(cid:13)L2 +(cid:13)(cid:13)W(cid:48)(u)(cid:13)(cid:13)L2
(cid:90) L2
(cid:105)(cid:17)

0

ε

as τ → 0.

|Lτ (z)|dz ≤ γ + O(τ ),

0,

L2
ε

that there exists a constant γ :=

R such that

which implies that Lτ ∈ L1(cid:16)(cid:104)
(cid:17)(cid:16)

Lτ →(cid:16)

z u − W(cid:48)(u)
∂2

(2.1.40)

Indeed our estimates applied to (2.1.25) show that

(cid:17)

z v − W(cid:48)(cid:48)(u)v
∂2

τ → 0

as

f or a.e

z,

(2.1.41)

Applying the Dominated convergence theorem, from (2.1.40) and (2.1.41) we show that the

limits

i(cid:48)(0) = lim
τ→0

i(τ ) − i(0)

τ

(cid:90) L2
(cid:16)

0

ε

= lim
τ→0

(cid:90) L2

ε

=

0

Lτ dz,

z u − W(cid:48)(u)
∂2

exist. This proves that i is differentiable at τ = 0.

23

(cid:17)(cid:16)

(cid:17)
z − W(cid:48)(cid:48)(u)
∂2

vdz,

(2.1.42)

2.2 The Euler-Lagrange Equation with a small mass

constraint

Recall that the inner scaling of the Functionalized Cahn-Hilliard free energy with no functionalization

terms is

(cid:90) L2

ε

0

1
2

(cid:16)

(cid:17)2
z u − W(cid:48)(u)
∂2

dz.

I (u) =

In this section, we address the dependence of the minima of the functional I upon the

mass M ,

(cid:90) L2

ε

0

(u − b−)dz = M,

(2.2.1)

by analyzing an asymptotic expansion of the Euler-Lagrange equation (2.1.10). We construct

a class of solutions of the E-L equation which corresponds to a scaling of the Lagrange

multiplier, λε = ελ, arising from the ε-dependence of the mass constraint, m. Following this

information, specifically, we seek solutions to the perturbed E-L equation,

2.2.1 Notation

Introduce the operator

(cid:16)

(cid:17)(cid:16)
z − W(cid:48)(cid:48)(Φ)
∂2

(cid:17)
z Φ − W(cid:48)(Φ)
∂2

= λε.

L := ∂2

z − W(cid:48)(cid:48)(φh),

24

(2.2.2)

(2.2.3)

that is the linearization of (1.2.5) about φh, and define the operator

Ln(p) := ∂2

z − W(cid:48)(cid:48) (un) ,

acting on H2 (R) where un is the extension φn to R defined as

n(cid:88)

(cid:0)z − pj

(cid:1) + b−,

(2.2.4)

(2.2.5)

un :=

φh

j=1

where φh := φh − b− and p = (p1, p2, ..., pn)t ∈ Rn is the vector of pulse locations. The

admissible set of pulse locations is given by

P = {p ∈ Rn : pi < pi+1

i = 1, ..., n and ∆p ≥ l},

for

(2.2.6)

|pi − pj| and l > 0 is sufficiently large. Let Bj denote the functions

where ∆p = min
i(cid:54)=j
L−j1 ∈ L∞(R) for j = 1, 2 that are the solutions to

LjBj = 1,

and orthogonal to the kernel of L. Actually, the function Bj is in the form

Specifically, B1 takes the form

Bj = ˆBj + Bj,∞.

B1 = ˆB1 − 1
α− ,

25

(2.2.7)

(2.2.8)

(2.2.9)

where α− is the coercivity value of W (u) at b− and ˆB1 is the solution to the

L ˆB1 = 1 − W(cid:48)(cid:48)(φh)
α− .

(2.2.10)

Since φh → b− at an exponential rate as z → ∞, the RHS of (2.2.10) is in L2(R) and
even about z = 0, hence orthogonal to φ(cid:48)
h ∈kerL. The existence of B2 is established
similarly. Further, we truncate Bj to have compact support on the bounded interval [0,
L2
ε ]

and introduce the functions

so that

n(cid:88)

i=1

Bj,n :=

ˆBj(z − pi) + Bj,∞,

Lj
n,bBj,n = 1 + o(δ2)

(2.2.11)

(2.2.12)

where Ln,b represents the restriction of the operator Ln(p) to the bounded domain [0,
Remark 2.2.1. The kernel of the operator L := ∂2
about the homoclinic solution φh is spanned by φ(cid:48)
h.

L2
ε ].
z − W(cid:48)(cid:48) (φh), the linearization of (1.2.6)

Indeed, the equation (1.2.6) has a

translational symmetry on R since φh(z + γ) also solves the equation (1.2.6). If we insert

φh(z + γ) in the ODE (1.2.6) and take its derivative with respect to γ we have

Lφh

(cid:48) = 0,

(2.2.13)

and conclude that kerL = span{φ(cid:48)

h}. Since L is a second order Sturm-Liouville operator

acting on an unbounded domain it has real-valued simple eigenvalues that can be written in a
(cid:48) is an eigenfunction of L corresponding
strictly decreasing order. Since kerL = span{φh

(cid:48)}, φh

26

to the eigenvalue λ = 0. Since the eigenfunctions of a Sturm-Liouville operator, {ψj}, has j
(cid:48) has one node, it is the second largest eigenfunction, ψ1, corresponding

simple zeros and φh

to the second eigenvalue λ1 = 0. Then, there exists a ground-state eigenfunction, ψ0
corresponding to λ0 > 0 and the remainder of the spectrum is real and O(1) distance

to the left of 0 by Weyl’s essential spectrum theorem.

Remark 2.2.2. By standard perturbation theory, the properties of the point spectrum of
Ln(p) presented in Lemma 4.1.4 carry over up to exponentially small terms to the operator
on the large bounded domain, Ln,b(p). (See Section 9.6 in [Kapitula and Promislow, 2013].)
For the purposes of exposition we do not distinguish between the operator Ln(p) acting
upon the whole line or Ln,b(p) acting upon the large bounded domain when inverting the

operator, except where doing so is essential to the argument.

2.2.2 Motivation

Before we proceed to the asymptotic expansion analysis of the E-L equation, we would like

to motivate ε-dependence of the Lagrange multiplier, λε, due to the ε-dependence of the

total mass, m. Assume uε is a solution to equation (2.2.2), or to

z u − W(cid:48)(cid:48)(cid:48)(u)(∂zu)2 +(cid:0)W(cid:48)(cid:48)(u)W(cid:48)(u) − λε

(cid:1) = 0,

(2.2.14)

z u − 2W(cid:48)(u)∂2
∂4

written explicitly.

Rather than dealing with possible different types of solutions to (2.2.2), we focus on

the construction of multipulse solutions as the possible minima of I. For this purpose, we
simplify the problem taking ε → 0 which extends the domain to R+ and the considering its

27

even extension to R−. Further, keeping the mass constraint fixed

(cid:90)

R

(u0 − b−)dz = M,

(2.2.15)

we deduce that u0 → b− as z → ±∞. Consistent with u0 → b− as z → ±∞, the exponential

dichotomies of (2.2.2) on R imply

z (u0 − b−) → 0 for k = 1, 2, 3 exponentially as
∂k

z → ±∞,

(2.2.16)

(See Section 2.1.4 in [Kapitula and Promislow, 2013]). In other words, there exist constants

c, κ > 0 such that

where (cid:126)b− = (b−, 0, 0, 0), (cid:126)u0(·) = (cid:0)u0(·), u(cid:48)

|(cid:126)u0 − (cid:126)b−| ≤ ce−κ|z|.

0 (·)(cid:1). Returning to (2.2.14), setting

(2.2.17)

0(·), u(cid:48)(cid:48)

0(·), u(cid:48)(cid:48)(cid:48)

ε = 0 and taking z → ±∞ by (2.2.16) we obtain the equality

W(cid:48)(b−)W(cid:48)(cid:48)(b−) = λ0.

(2.2.18)

Since W(cid:48)(b−) = 0 we conclude that λ0 = 0.

For the finite domain problem associated to (2.2.2), namely when ε (cid:54)= 0 but small, we

choose boundary conditions that best approximate the whole line problem. In particular,

we assume u becomes asymptotically close to b which solves

W(cid:48)(b)W(cid:48)(cid:48)(b) = λε,

(2.2.19)

28

and impose the exponential dichotomies like boundary conditions

(cid:126)u(0) ∈ Wu((cid:126)b) and (cid:126)u

∈ Ws((cid:126)b),

where (cid:126)b = (b, 0, 0, 0), (cid:126)u(·) = (cid:0)u(·), u(cid:48)(·), u(cid:48)(cid:48)(·), u(cid:48)(cid:48)(cid:48)(·)(cid:1) and Wu((cid:126)b) and Ws((cid:126)b) are the

(cid:18) L2

(cid:19)

ε

unstable and stable manifolds of (cid:126)b, respectively. This yields a finite domain problem that

best approximates the whole-line problem. This assumption is consistent with the mass

constraint, away from the pulses scale like

(cid:90) L2

ε

0

M =

(u − b−)dz ≈ L2
ε

(b − b−) = O(1),

(2.2.20)

and we deduce that b = b− + O(ε). From (2.2.19) we deduce that

λε =(cid:0)W(cid:48)(cid:48)(b−)(cid:1))2(b − b−) = O(ε).

(2.2.21)

To fix notation, we write b = b− + εb1 and λε = ελ for b1 = O(1) and λ = O(1).

2.2.3 Solutions to the Euler-Lagrange Equation with a small mass

constraint

In the light of the comments about our motivation discussed in Section 2.2.2, our main focus

of interest is constructing a class of solutions to (2.2.2) when λ
 = 
λ. We are specifically

interested in an asymptotic expansion of these solutions which has an expansion form

Φ = ϕ0 + εϕ1 + O(ε2).

(2.2.22)

29

(a) Phase plane for when λ
 = 0

(b) Phase plane when λ
 (cid:54)= 0

Figure 2.1: Figure 2.1a represents a phase plane for the dynamical system given in (2.2.2)
extended to R for λ
 = 0. Figure 2.1b is a phase plane for the dynamical system when
λ
 (cid:54)= 0. The boundary conditions (cid:126)u(0) ∈ W u((cid:126)b) and (cid:126)u
mimic the behavior of the whole line system. The distance between the fixed points of the
dynamical systems is |b− b−| = ε while |u(0)− b| = O(e−κ
L2
ε ) (cid:28) ε. However, for the clarity
of the graph both distances are depicted in similar lengths.

(cid:17) ∈ W s((cid:126)b) where (cid:126)b = (b, 0, 0, 0)

(cid:16) L2

ε

(cid:16)
(cid:16)
(cid:16)

ελ =

=

=

To find ϕ0 and ϕ1 we first insert (2.2.22) in (2.2.2).

(cid:17)(cid:16)
(cid:17)
z Φ − W(cid:48) (Φ)
z − W(cid:48)(cid:48) (Φ)
(cid:17)(cid:17)(cid:16)
∂2
∂2
ϕ0 + εϕ1 + O(ε2)
∂2

z − W(cid:48)(cid:48)(cid:16)
(cid:16)
z −(cid:0)W(cid:48)(cid:48) (ϕ0) + εϕ1W(cid:48)(cid:48)(cid:48) (ϕ0+)(cid:1) + O(ε2)

∂2
z

∂2

,

(cid:17)(cid:16)

ϕ0 + εϕ1 + O(ε2)
(cid:17)
−(cid:0)W(cid:48) (ϕ0) + εϕ1W(cid:48)(cid:48) (ϕ0)(cid:1) + O(ε2)

∂2
z ϕ0 + ε∂2

.

(cid:17) − W(cid:48)(cid:16)

(cid:17)(cid:17)

,

ϕ0 + εϕ1 + O(ε2)

z ϕ1+

(2.2.23)

Matching the terms with the same order of ε, we obtain O(1) and O(ε) equations

(cid:16)

(cid:17)(cid:16)

(cid:17)
z ϕ0 − W(cid:48) (ϕ0)
∂2

z − W(cid:48)(cid:48) (ϕ0)
∂2

= 0,

(2.2.24)

30

(cid:16)

z − W(cid:48)(cid:48) (ϕ0)
∂2

(cid:17)(cid:16)

z ϕ1 − ϕ1W(cid:48)(cid:48) (ϕ0)
∂2

(cid:17)

(cid:16)

(cid:17)
z ϕ0 − W(cid:48) (ϕ0)
∂2

+ ϕ1W(cid:48)(cid:48)(cid:48) (ϕ0)

= λ,

(2.2.25)

As our base profile we choose the solutions φn for n ∈ { k+1

2

: k ∈ N} to the problem (1.2.5)

as a special classes of solutions of equation (2.2.24). Inserting ϕ0 = φn in (2.2.25),

λ =

(cid:16)
(cid:16)

(cid:17)(cid:16)
z − W(cid:48)(cid:48) (φn)
(cid:17)(cid:16)
∂2
z − W(cid:48)(cid:48) (φn)
∂2
=
= L2

nϕ1,

(cid:17)

z ϕ1 − ϕ1W(cid:48)(cid:48) (φn)
(cid:17)
∂2
z − W(cid:48)(cid:48) (φn)
∂2

ϕ1,

,

(2.2.26)

where we have denoted the linearization of (1.2.6) about the periodic solution φn by

For each n = 1, 2, 3, the equation

Ln := ∂2

z − W(cid:48)(cid:48)(φn).

L2
nϕ1 = λ,

(2.2.27)

(2.2.28)

has a solution ϕ1 ∈ L2 if and only if λ ⊥ kerLn by the Fredholm Alternative. Indeed, λ is
orthogonal to kerLn because we have

(cid:90)

R

(cid:48)dz = 0,

λφn

(2.2.29)

since kerLn = span{φn

(cid:48)} and φn

(cid:48) is odd about z = 0. Then, the solution to (2.2.28) can be

written as

ϕ1 = λB2,n,

(2.2.30)

31

where B2,n solves

L2
nB2,n = 1.

(2.2.31)

Therefore, the periodic solutions to the Euler-Lagrange equation (2.1.10) in other words the

critical points of I subject to the mass constraint (1.2.4) have the asymptotic expansion

Φn = φn + ελB2,n + O(cid:16)

ε2(cid:17)

.

(2.2.32)

2.2.4 Some Remarks

Recall that the equation (1.2.5) has solutions, φn, which are global minimizers of the free

energy (1.2.3) when I(u) is not subject to any constraints. These have background state,

b = b−. In Section 2.2.2 we motivate the construction of the solutions with background.
b = bn where bn − b− = O(ε) solves

W(cid:48)(bn)W(cid:48)(cid:48)(bn) = λε.

(2.2.33)

Since W(cid:48)(b−) = 0 it follows λε is also O(ε). Let τ (bn) be the period of the orbit φn. The

periodic solutions Φn to the Euler-Lagrange equation (2.1.10) which satisfy τ (bn) =

fit precisely n periods of the orbit into the interval

solve the boundary conditions. Moreover, the period scales like τ (bn) = O(cid:16)

and
is monotonically decreasing as b−b− increases, achieving a minima at the center equilibrium.

and may be translated to exactly

(cid:16) 1

(cid:17)(cid:17)

bn−b−

ln

L2
nε can

, however we further restrict the size

(cid:105)

(cid:104)

0,

L2
ε

ετ (bn) = O(cid:16) 1

L2

ε|lnε|

(cid:17)

32

These considerations suggest n =
of bn so that n = O(1)
Remark 2.2.3. If τ (b) (cid:54)=

L2
nε then we state without proof that by adjusting the associated φ

for which (φ, φ(cid:48)) passes through (b, 0), we may arrive at a solution of (1.2.5) that satisfies
ε ) = O(e− L2
the boundary conditions. Indeed by translating φ we may achieve φ(cid:48)(0), φ(cid:48)(

ε ).

L2

The corrections to φ are exponentially small and have no impact on the value of the energy

or total mass to the order considered here. In particular, we define

(cid:22) L2

τ (b)ε

(cid:23)

− 1
2

n :=

.

(2.2.34)

we proceed formally with the construction of Φn ignoring the issue of exponentially small

boundary mismatch.

2.2.5 Energy values at the critical points

Inserting the critical points (2.2.32) in (1.2.3) and in the mass constraint (1.2.4) we calculate

their energy values and their mass constraint M to determine which has minimum energy

at prescribed mass.

0

(cid:90) L2
(cid:90) L2
(cid:90) L2
(cid:90) L2
(cid:90) L2

0

0

0

ε

0

ε

ε

ε

ε

∂2
z

(cid:17)2

dz,

φn + ελB2,n + O(ε2)

(cid:16)
z Φn − W(cid:48) (Φn)
∂2
(cid:16)
(cid:16)
ε2λ2(cid:16)
ε2λ2(cid:0)LnB2,n
ε2λ2(cid:0)LnB2,n

zB2,n − W(cid:48)(cid:48) (φn)B2,n
∂2
(cid:1)2 + O(ε3)dz,
(cid:1)(cid:0)LnB2,n

1
2

1
2

1
2

1
2

1
2

I (Φn) =

=

=

=

=

(cid:1) + O(ε3)dz.

(cid:17)(cid:17)2
φn + ελB2,n + O(ε2)

dz,

(cid:17) − W(cid:48)(cid:16)
(cid:17)2

+ O(ε3)dz,

(2.2.35)

33

Since Ln’s are self-adjoint and from (2.2.31) we obtain the reduced energy

(cid:90) L2
(cid:90) L2

0

ε

ε

0

1
2

1
2

I (Φn) =

=

ε2λ2(cid:16)L2

nB2,n

(cid:17)B2,n + O(ε3)dz,

ε2λ2B2,n + O(ε3)dz.

Let ¯B2,n denote the mass for B2,n that is
(cid:90) L2

B2,n :=

ε

B2,ndz,

0

= α−2− L2

ε

+ O(1),

and Mn be the mass for the critical point (2.2.32)

(cid:90) L2
(cid:90) L2

0

ε

ε

0

Mn : =

=

Φndz

(cid:2)(φn − b−) + ελB2,n

(cid:3) dz.

(2.2.36)

(2.2.37)

(2.2.38)

Substituting the mass (2.2.37), we have

Mn = L2α−2− λ +

(cid:90) L2

ε

0

(φn − b−)dz.

(2.2.39)

For the sake of convenience, we write the mass of φn in terms of the mass of φh,

(φn − b−)dz = n

(φh − b−)dz,

(2.2.40)

(cid:90) L2

ε

0

(cid:90) L2

ε

0

= nMh,

34

where Mh is the mass for the homoclinic solution φh, namely

(cid:90) L2

ε

0

Mh :=

(φh − b−)dz.

We insert (2.2.40) in (2.2.38) and obtain

Mn = nMh + λα−2− L2.

(2.2.41)

(2.2.42)

Setting Mn = M and solving for λ gives an expression for λ in terms of the mass

constraint M ,

λ =

α2−
L2

(M − nMh) .

(2.2.43)

Utilizing this formula of λ in (2.2.36), we calculate the reduced energy critical point of I in

(cid:90) L2

ε

terms of the mass constraint M

I (Φn) =

=

=

ε2λ2B2,ndz,

1
2

0
ε2λ2B2,n,
1
2
εα2−
2L2

(M − nMh)2 .

(2.2.44)

We minimize (2.2.44) for n to find the n-pulses with the minimal free energy. Note that

when M = nMh, Φn is a global minimizer. Since I(Φn) is a discrete function of n, the

closest value of n to M
Mh

is the minima of I (Φn). We conclude that the inner scaling of FCH

free energy, I(u), attains its minima, over the n pulse solutions we have constructed, at Φn

for which n is closest to M
Mh

.

From Figure 2.2 , it can be observed that among the n-pulse profiles we have constructed

35

Figure 2.2: The reduced energy, I(Φn), versus mass constraint M . The blue lines
demonstrate the energy values at the critical points, Φn, for n = 0, 1
2 , 2. The red
lines represent the infimum of the energy values over all the blue lines.

2 , 1, 3

the one with minimal energy sensitively depend on the mass constraint. In the sequel, we

show the existence of a global minima and that integral values of n the associated n-pulse

Φn is a local minima of I.

36

01/2mhmh3/2mh2mhM*MI(n)I(0)(M)I(1/2)(M)I(1)(M)I(3/2)(M)I(2)(M)1/23/23/22BAGH--←izµ←Chapter 3

Existence Of the Minimizers

3.1 Existence of the Minimizers

In this section, we use classical tools from direct methods of Calculus of Variations to

establish the existence of a global minimizer the energy functional (1.2.3) subject to the

mass constraint (1.2.4) following the procedures in [Promislow and Zhang, 2013]. Recall

that the space of admissible functions is given by

(cid:26)

(cid:18)(cid:20)

(cid:21)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) L2

ε

0

0,

L2
ε

A =

u ∈ H2

(u − b−)dz = M, uz(0) = 0, uz

(cid:18) L2

(cid:19)

ε

(cid:27)

= 0

.

We consider a general form for the double-well potential W (u).

In addition to the

assumptions earlier on, we suppose that W (u) is convex at infinity and satisfies some growth

conditions and b− = 0 for the sake of easiness in the calculations. Specifically, there exist

p > 1, c, c− > 0, u0 > 0 and β > 1 sufficiently large such that

 W(cid:48)(u) ≤ c|u|p,

W(cid:48)(u) ≤ c−,

|u| ≥ 1,

|u| < 1,

W(cid:48)(cid:48)(u) > β,

|u| > u0.

37

(3.1.1)

(3.1.2)

In the following lemma, we establish an estimate for H1 norm of u ∈ A which is utilized in

the proof of the H2-coercivity of the energy in Theorem 3.1.2.

For simplicity in the proof of existence of a global minima, for fixed ε > 0 and L2 ∈ R we

introduce scaled Cahn-Hilliard free energy functional on a large bounded domain

0,

(cid:104)

(cid:105) ∈ R

L2
ε

(cid:90) L2

ε

0

1
2

E =

(∂zu)2 + W (u)dz.

(3.1.3)

The variational derivative of E with respect to L2 inner product is

= −∂2

z u + W(cid:48)(u),

δE
δu

and note that the FCH free energy, I, can be written in terms of δE
δu

(3.1.4)

(3.1.5)

I(u) =

=

(cid:19)2

dz,

ε

.

δu

δu

1
2

0
1
2

(cid:18) δE
(cid:90) L2
(cid:13)(cid:13)(cid:13)(cid:13)δE
(cid:13)(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)(cid:13) δE

L2

δu

L2

Lemma 3.1.1. Fix ε > 0. There exists a constant C > 0 such that

(cid:107)u(cid:107)2

H1 ≤

+ C,

(3.1.6)

for all u ∈ A.

Proof. Multiplying δE

δu by u and integrating by parts, we obtain

(cid:90) L2

ε

0

(cid:90) L2

ε

0

(cid:16)−∂2

(cid:17)
z u + W(cid:48)(u)

u

dz =

(cid:90) L2

ε

0

u

δE
δu

dz =

38

(∂zu)2 + uW(cid:48)(u)dz.

(3.1.7)

So,

(cid:90) L2

ε

0

(cid:90) L2

ε

0

(∂zu)2dz =

− uW(cid:48)(u)dz.

u

δE
δu

(3.1.8)

Applying Young’s inequality to the first term on the right and adding (cid:107)u(cid:107)2

L2 to both

sides, we have

(cid:90) L2

ε

0

(cid:107)u(cid:107)2

H1 ≤

(cid:18) δE

(cid:19)2 − uW(cid:48)(u) + u2dz,

δu
u2 − uW(cid:48)(u)dz.

= I(u) +

u2
2

1
2

ε

+

(cid:90) L2
(cid:90) L2

0

3
2

ε

We need to get an upper bound for

h(z)dz where

0

h(z) :=

u2 − uW(cid:48)(u).

3
2

From the assumption (3.1.2) there exists β and u0 such that

W(cid:48)(u) > β (u − u0) + W(cid:48)(u0)

for u > u0,

W(cid:48)(u) < β(u0 + u) + W(cid:48)(−u0)

for u < −u0.

Multiplying (3.1.11) and (3.1.12) by −u and adding 3

2 u2 , we obtain

(3.1.9)

(3.1.10)

(3.1.11)

(3.1.12)

(cid:19)

− β

(cid:19)

(cid:18)3
(cid:18)3

2

2

h(u) <

u2 + βu0u − uW(cid:48)(u0)

for u > u0,

(3.1.13)

h(u) <

− β

u2 − βu0u − uW(cid:48)(−u0)

for u < −u0.

(3.1.14)

The inequalities (3.1.13) and (3.1.14) imply that on R/[−u0, u0] h(u) is bounded above

for sufficiently large β because it is bounded by a function whose dominant term is quadratic

39

with a negative coefficient for sufficiently large β. Further, h(u) is bounded on [−u0, u0]

since it is continuous. Hence, h(u) is bounded above for sufficiently large β, in other words

there exists a constant C > 0 independent of ε such that

(cid:90) L2

ε

0

h(z)dz < C.

(3.1.15)

From this and (3.1.9) , the bound we aimed in (3.1.6) has been achieved.

Theorem 3.1.2. Fix ε > 0 and L2, independent of 
. The energy functional given in (1.2.3)
subject to the mass constraint (1.2.4) has a global minimizer over the admissible set A. In
other words, there exists at least one u ∈ A satisfying

I(w) ≥ I(u),

(3.1.16)

for all w ∈ A.

Proof. Since I(·) is well-defined on A and bounded below by 0, we define m := infw∈AI(w) ≥

0.

The key step in the proof of the existence of a minimizer is to show the coercivity of I(u)
over the admissible set A. More specifically we show that there exist constants β and γ such

that

(I(u))

3

2 ≤ −γ + β(cid:107)u(cid:107)H2.

(3.1.17)

To establish this bound we pursue the preliminary step of bounding the variational derivative

40

of E, given in (3.1.3). By the relation obtained from (3.1.4)

∂2
z u =

δE
δu

− W(cid:48)(u).

Taking L2 norm of both sides and applying the triangle inequality, we obtain

(cid:13)(cid:13)(cid:13)∂2

z u

(cid:13)(cid:13)(cid:13)L2

δu

(cid:13)(cid:13)(cid:13)(cid:13)δE
(cid:13)(cid:13)(cid:13)(cid:13)δE
(cid:13)(cid:13)(cid:13)(cid:13)δE

δu

δu

,

− W(cid:48)(u)

(cid:13)(cid:13)(cid:13)(cid:13)L2
+(cid:13)(cid:13)W(cid:48)(u)(cid:13)(cid:13)L2 ,

(cid:13)(cid:13)(cid:13)(cid:13)L2
(cid:13)(cid:13)(cid:13)(cid:13)L2
+(cid:13)(cid:13)W(cid:48)(u)(cid:13)(cid:13)L2 ,
2 +(cid:13)(cid:13)W(cid:48)(u)(cid:13)(cid:13)L2 .

1

=

≤

≤

= (2I(u))

and by the second relation in (3.1.5),

(cid:13)(cid:13)(cid:13)∂2

z u

(cid:13)(cid:13)(cid:13)L2

Then, we add (cid:107)u(cid:107)H1 on both sides of (3.1.20) and obtain

(cid:107)u(cid:107)H2 ≤ (2I(u))

2 +(cid:13)(cid:13)W(cid:48)(u)(cid:13)(cid:13)L2 + (cid:107)u(cid:107)H1 .

1

Here using the bound for H1 norm of u constructed in Lemma 3.1.1, we have

(cid:107)u(cid:107)H2 ≤ (2I(u))

≤ (2I(u))

1

2 +(cid:13)(cid:13)W(cid:48)(u)(cid:13)(cid:13)L2 + (cid:107)u(cid:107)H1 ,
(cid:32)(cid:13)(cid:13)(cid:13)(cid:13)δE
(cid:13)(cid:13)(cid:13)(cid:13)2
2 +(cid:13)(cid:13)W(cid:48)(u)(cid:13)(cid:13)L2 +

δu

1

L2

(cid:33) 1

2

+ C

,

and by the second relation in (2.1.24), we obtain

(cid:107)u(cid:107)H2 ≤ (2I(u))

2 +(cid:13)(cid:13)W(cid:48)(u)(cid:13)(cid:13)L2 + (2I(u) + C)

1

1
2 .

41

(3.1.18)

(3.1.19)

(3.1.20)

(3.1.21)

(3.1.22)

(3.1.23)

Further, by the assumptions (3.1.1) we have

(cid:13)(cid:13)W(cid:48)(u)(cid:13)(cid:13)2

L2 = c2

(cid:90)
(cid:90) L2

ε

|u|>1

(cid:90)
|u|2pdz + c2−
(cid:90)

|u|2pdz + c2−

≤ c2

0

|u|2dz,

|u|<1

|u|2dz,

(3.1.24)

|u|<1

≤ c2 (cid:107)u(cid:107)2p
≤ c2 (cid:107)u(cid:107)2p

L2p + c2− L2
H1 + c2− L2

ε

ε

.

Taking square root of both sides of (3.1.24) and by Lemma 3.1.1, we deduce that there exists

a c > 0 such that

(cid:13)(cid:13)W(cid:48)(u)(cid:13)(cid:13)L2 ≤ c

(cid:18)
(cid:18)

1 +

(cid:13)(cid:13)(cid:13)(cid:13)δE

δu

(cid:13)(cid:13)(cid:13)(cid:13)p

(cid:19)
(cid:19)

= c

1 + (2I(u))

,

,

(3.1.25)

L2
p
2

for p > 1. Inserting (3.1.25) in (3.1.23) we conclude that there exist constants α, γ > 0 such

that

(cid:107)u(cid:107)H2 ≤ α + γ (I(u))

p
2 ,

(3.1.26)

for p > 1 and this provides the H2 coercivity of I(u).

The other essential part of the proof is showing the weak lower semi-continuity of the

energy. Since the energy functional is bounded below, there exists a minimizing sequence

{uk}∞

k=1 ∈ A and so,

I(uk) → m.

(3.1.27)

From the H2 coercivity of I(u), the sequence {uk} is bounded in H2 and there exists a

42

subsequence {ujk

} and ¯u ∈ H2 such that

(cid:42) ¯u weakly in H2,

ujk

(3.1.28)

)n → ¯u in H1 since H1 ⊂⊂ H2. Before the
and further, there exists a subsequence (ujk
proof of weak lower semi-continuity of I(u), we need to verify that such ¯u resides in A. First

(cid:42) ∂z ¯u weakly in L2 and ∂2

(cid:42) ∂2

z ¯u in L2. From integration by parts,

z ujk

observe that ∂zujk

for any w ∈ H2(cid:16)(cid:104)

0,

for any v ∈ C∞(cid:16)(cid:104)

0,

we have

(cid:17)

(cid:16)

∂2
z w

vdz = −

(cid:90) L2

ε

0

∂zw∂zv + (∂zw) v(cid:12)(cid:12) L2

ε

0

,

(3.1.29)

. We substitute w = ujk

in (3.1.31).Since ujk

∈ A the boundary

L2
ε

(cid:105)(cid:17)
(cid:90) L2
(cid:105)(cid:17)

0

ε

L2
ε

ε

(cid:90) L2
(cid:105)(cid:17)

0

term is 0 and hence the equation (3.1.31) becomes

(cid:90) L2

ε

(cid:16)

(cid:17)

∂2
z ujk

vdz = −

ε

(cid:90) L2
(cid:105)(cid:17)

0

0,

L2
ε

0

On the other hand, we substitute w = ¯u ∈ H2(cid:16)(cid:104)
(cid:90) L2

(cid:17)

(cid:16)

∂2
z ¯u

vdz = −

ε

0

for all v ∈ C∞(cid:16)(cid:104)

0,

L2
ε

∂zujk

∂zv.

(3.1.30)

in (3.1.31) and obtain

∂z ¯u∂zv + (∂z ¯u) v(cid:12)(cid:12) L2

ε

0

,

(3.1.31)

. Consequently, since ∂zujk

(cid:42) ∂z ¯u weakly in L2 and ∂2

z ujk

(cid:42) ∂2
z ¯u

in L2 and comparing the non-zero terms in (3.1.30) and (3.1.31) we conclude that

(∂z ¯u) v(cid:12)(cid:12) L2

ε
0 = 0,

43

(3.1.32)

for all v ∈ C∞(cid:16)(cid:104)

(cid:105)(cid:17)

0,

L2
ε

. This proves that ∂z ¯u(0) = ∂z ¯u(

ε ) and hence, ¯u ∈ A.
L2

For the weak lower-semi continuity of I(u), we utilize the weak convergence of the first

variation of F,

(cid:16)

δF
δu

Dujk

, ujk

, z

(cid:17)

δF
δu

(cid:42)

which is established in Lemma 3.1.3.

(D¯u, ¯u, z) weakly in L2,

(3.1.33)

Since weak convergence is lower semi continuous,

(cid:90) L2

ε

(cid:18) δF

0

δu

(cid:16)

Dujk

, ujk

, z

(cid:17)(cid:19)2

dz ≥

(cid:90) L2

ε

0

(cid:18) δF

δu

(cid:19)2

(D¯u, ¯u, z)

dz.

lim inf

Considering ¯u ∈ A, it follows that

m ≤ I(¯u) ≤ lim inf I(ukj

) = m.

Thus, the energy, I, attains its minima at ¯u.

(3.1.34)

(3.1.35)

(cid:42) ¯u weakly in H2 and strongly in L2, then the variational derivative

Lemma 3.1.3. If ujk
of F, given in (3.1.3), converges

δF
δu

(cid:105)(cid:17)

.

0,

L2
ε

weakly in L2(cid:16)(cid:104)
W(cid:48)(cid:16)

ujk

Proof. We already know that ∂2

(cid:17) → W(cid:48)(¯u) in L2(cid:16)(cid:104)

z ujk
L2
ε

0,

uniformly bounded there exists ξjk

(cid:16)

(cid:17)

δF
δu

(cid:42)

Dujk

, ujk

, z

(D¯u, ¯u, z) ,

(3.1.36)

(cid:42) ∂2

z ¯u in L2(cid:16)(cid:104)
(cid:105)(cid:17)
(cid:105)(cid:17)
∈ L∞(cid:16)(cid:104)

0,

L2
ε

(cid:105)(cid:17)
(cid:13)(cid:13)(cid:13)ξjk

0,

L2
ε

with

44

. By the mean value theorem and since ¯u, ujk

and it suffices to show that

(cid:13)(cid:13)(cid:13)L∞ uniformly bounded,

are

independent of ε, such that

(cid:12)(cid:12)W(cid:48)(cid:16)

ujk

Squaring both sides of (3.1.37) and integrating over

0,

(cid:13)(cid:13)(cid:13)W(cid:48)(cid:16)

ujk

(cid:13)(cid:13)(cid:13)L2
(cid:17) − W(cid:48)(¯u)

weakly in H2(cid:16)(cid:104)
which converges to zero since W(cid:48)(cid:48)(ξjk
in L2(cid:16)(cid:104)
(cid:105)(cid:17)
and also, ujk

(cid:105)(cid:17)

L2
ε

0,

0,

.

L2
ε

L2
ε

− ¯u|.

we obtain

)||ujk
(cid:105)

(cid:17) − W(cid:48)(¯u)(cid:12)(cid:12) ≤ |W(cid:48)(cid:48)(ξjk
(cid:104)
≤(cid:13)(cid:13)(cid:13)W(cid:48)(cid:48)(ξjk
≤(cid:13)(cid:13)(cid:13)W(cid:48)(cid:48)(ξjk
) is uniformly bounded in L∞(cid:16)(cid:104)
→ ¯u in L2(cid:16)(cid:104)
. Thus, W(cid:48)(ujk

(cid:13)(cid:13)(cid:13)L2
(cid:13)(cid:13)(cid:13)ujk

(cid:13)(cid:13)(cid:13)L∞

− ¯u (cid:107)L2,

(cid:105)(cid:17)

− ¯u)

)(ujk

0,

L2
ε

)

,

(3.1.37)

(3.1.38)

(cid:105)(cid:17)

0,

and ujk

L2
(cid:42) ¯u
ε
) converges W(cid:48)(¯u)

45

Chapter 4

Modulational Stability of n-pulses

In this chapter, our main interest is the dynamical stability of a manifold of n-pulses given

as the graph of an n-pulse ansatz, Φn(z, p, ¯λ). These are related to the periodic multi-pulse
solutions, {Φn|n ∈ N}, constructed in Chapter 2 as a class of critical points to the free
energy I. The H2-coercivity of the second variation of I about Φn(z, p, ¯λ), modulational

stability and dynamic evolution of the n-pulse ansatz with respect to the Π0-gradient flow

are addressed.

4.1 H 2-coercivity of the second variation of I

In this section, we prove the H2-coercivity of the second variation of the free energy I

about n-pulse ansatz,

representation theorem

δ2I
δu2 (Φn(z, p), ¯λ). The second variation is defined from the Riesz

(cid:12)(cid:12)(cid:12)τ =0

d2
dτ 2 I(u + τ v)

(cid:68) δ2I

δu2 (u)v, v

(cid:69)

,

L2

=

(4.1.1)

for any v ∈ H2(R).

Definition 4.1.1. : Let D be a subspace of Hilbert space H. A linear operator A : D → H

satisfying

(cid:104)Au, u(cid:105) ≥ µ(cid:107)u(cid:107)2 ,

∀u ∈ D

(4.1.2)

46

for some µ > 0 is called a coercive operator and µ is called the coercivity constant.

4.1.1 Introduction: n-pulses

Recall that n-pulse ansatz defined on R is given in (2.2.5) as

n(cid:88)

(cid:0)z − pj

(cid:1) + b−,

φh

(4.1.3)

un :=

j=1

where φh := φh − b− and p = (p1, p2, ..., pn)t ∈ Rn is the vector of pulse locations. The

admissible set of pulse locations is given by

P = {p ∈ Rn : pi < pi+1

i = 1, ..., n and ∆p ≥ l},

for

(4.1.4)

where ∆p = min
i(cid:54)=j

|pi−pj| and l > 0 is sufficiently large so that the exponential terms e−√

α−l

arising in the calculations are negligible. We extend un to be defined on all of R, and add
a correction term that reduces the size of residual. Recalling the definition of B2,n given in

(2.2.31), we introduce the corrected extension by

Φn(z, p, ¯λ) := un + δ¯λB2,n,

(4.1.5)

and let Mn = {Φn(p, ¯λ)|p ∈ P} be the n-dimensional manifold formed by these solutions.
Let Xn(p) represent the tangent plane to the manifold Mn, Xn(p) = span{ ∂Φn(p)
: p ∈
∂pi
P ⊂ Rn} = span{φ(cid:48)

h(z − pi)}n

i=1.

Recall that in (2.2.4) we introduced the operator

Ln(p) := ∂2

z − W(cid:48)(cid:48) (un) ,

47

(4.1.6)

acting on H2 (R). The second variation of the free energy I about n-pulse ansatz is

(cid:16)

L :=

z − W(cid:48)(cid:48)(Φn)
∂2

(cid:17)2 −(cid:16)

(cid:17)
z Φn − W(cid:48)(Φn)
∂2

W(cid:48)(cid:48)(cid:48)(Φn).

(4.1.7)

Taylor expanding (4.1.7) about un up to O(δ2) terms we obtain

L =

(cid:17)2 −(cid:16)

(cid:16)Ln − δ¯λW(cid:48)(cid:48)(cid:48)(un)B2,n + O(δ2)
(cid:17)(cid:16)
− δ¯λW(cid:48)(cid:48)(un)B2,n + O(δ2)
+ O(δ2)

(cid:17)

.

z un − W(cid:48)(un) + δ¯λ∂2
∂2
W(cid:48)(cid:48)(cid:48)(un) + δ¯λW (4)(un)B2,n+

zB2,n

(4.1.8)

(4.1.9)

.

Recalling that LnB2,n = B1,n we obtain

(cid:16)Ln − δ¯λW(cid:48)(cid:48)(cid:48)(un)B2,n + O(δ2)

(cid:17)2 −(cid:16)

L =

z un − W(cid:48)(un)+
∂2

(cid:17)
W(cid:48)(cid:48)(cid:48)(un) + δ¯λW (4)(un)B2,n + O(δ2)

(cid:17)(cid:16)
+ δ¯λB1,n + O(δ2)
(cid:105)
for i = 1, . . . , n we have un = φi +O(e−√
z un − W(cid:48)(un) about φh(z − pi) and obtain ∂2

From the definition of un, partitioning the domain into sub-intervals about pulse locations as
α−(cid:96)) on each sub-interval. We
z un − W(cid:48)(un) = −W(cid:48)(cid:48)(φh(z −

Taylor expand ∂2

pi+pi+1

(cid:104) pi−1+pi

2

2

,

pi))e−√

√
α−(cid:96) + O(e−2

α−(cid:96)) on

Summing up over all sub-intervals provides ∂2
Here we set δ = e−√

α−(cid:96).
α−(cid:96) which will be our scaling value of the background parameter through

i=1

W(cid:48)(cid:48)(φh(z − pi))e−√

since φh solves the equation (1.2.6).

(cid:104) pi−1+pi

2

pi+pi+1

,

(cid:105)
z un − W(cid:48)(un) = − n(cid:80)

2

the rest of this thesis.

48

We insert this expansion into (4.1.9) and calculate the inner product

(cid:17)

W(cid:48)(cid:48)(cid:48)(un)+

(4.1.10)

(cid:104)Lv, v(cid:105) =

(cid:68)(cid:18)(cid:16)Ln − δ¯λW(cid:48)(cid:48)(cid:48)(un)B2,n + O(δ2)
+ −(cid:16)−δ
n(cid:88)
(cid:19)

(cid:17)2
W(cid:48)(cid:48)(φh(z − pi)) + δ¯λB1,n
(cid:69)

i=1

+

+ O(δ2)

v, v

,

for any v ∈ H2. Here we must be careful when expanding the quadratic term and simplifying

further. Precisely, we have

(cid:104)Lv, v(cid:105) =

(cid:68)(cid:16)L2
nv − δ¯λLn(W(cid:48)(cid:48)(cid:48)(un)B2,nv) − δ¯λW(cid:48)(cid:48)(cid:48)(un)B2,nLnv
+ −(cid:16)−δ
W(cid:48)(cid:48)(cid:48)(un)v+
(cid:69)

W(cid:48)(cid:48)(φh(z − pi)) + δ¯λB1,n

n(cid:88)

(cid:17)

i=1

v, v

+ O(δ2 (cid:107)v(cid:107)2

H2).

(cid:17)

+

(4.1.11)

δ2I
Further, for the proof of H2-coercivity of L and later to verify H2-coercivity of
δu2 (φn) :=
L2
n we establish and utilize H2-coercivity of the second variation of the inner scaling of FCH

free energy about n-pulse ansatz at the leading order so it is worth noting that

(cid:104)Lv, v(cid:105) = (cid:104)L2

nv, v(cid:105) + O(δ (cid:107)v(cid:107)2

H2).

(4.1.12)

4.1.2 H 2-coercivity of the second variation of I about n-pulse ansatz

For the stability analysis of the n-pulse ansatz purposes, we establish the H2-coercivity of

the second variation of I about n-pulse ansatz given in (4.1.10).

Theorem 4.1.2. Consider the inner scaling of FCH free energy, I, given in (1.2.3) and

49

n-pulse ansatz given in (4.1.5). Then, the bilinear form induced by L given in (4.1.10) is

coercive, i.e, there exists a µ > 0 independent of ε such that

(cid:104)Lv, v(cid:105) ≥ µ(cid:107)v(cid:107)H2(R),

(4.1.13)

for all v ∈ X⊥

n (p).

The essential step in the proof of this theorem is the H2-coercivity of L2

n over X⊥
Before we present the proof of Theorem 4.1.2 we establish the H2-coercivity of L2
X⊥
n (p).

n (p).

n over

Theorem 4.1.3. Consider the operator Ln given in (2.2.4) on H2 (R) which is the linearization

of (1.2.6) about n-pulses, un, given in (2.2.5). Then, the bilinear form induced by
L2
n is coercive, i.e, there exists a ˜µ > 0 independent of ε such that

(cid:104)L2

nv, v(cid:105) ≥ ˜µ(cid:107)v(cid:107)H2(R),

for all v ∈ X⊥

n (p).

δ2I
δu2 (un) =

(4.1.14)

The H2-coercivity of the operator L2

n(p) arises from the L2-coercivity of L2

n(p). We

prove the L2-coercivity of L2

n(p) in Lemma 4.1.4.

Lemma 4.1.4. There exists a ˜µ > 0 such that

(cid:104)L2

n(p)v, v(cid:105) ≥ ˜µ(cid:107)v(cid:107)2

L2(R)

,

(4.1.15)

for all v ∈ X⊥

n (p).

50

Proof. It suffices to show that there exists a ˜µ > 0 such that

(cid:104)(L2

n(p) − ˜µ)v, v(cid:105) ≥ 0,

(4.1.16)

for all v ∈ X⊥

n (p). Ln

2

(p) − ˜µ : H2(R) ⊂ L2(R) → H−2(R) is a self-adjoint operator on

L2(R). Let

b[v, v] := (cid:104)(L2

n(p) − ˜µ)v, v(cid:105),

(4.1.17)

be the bilinear form associated to Ln
induced by the constrained bilinear form defined by the restriction of b[v, v] to X⊥

(p) in the L2 inner product. We first find the operator

2

n (p).

We introduce the orthogonal projection P : H2(R) → Xn(p) and Π := I − P which has

the range X⊥

n (p). The bilinear form constrained to X⊥

n (p) induces the constrained operator

(L2

n(p) − ˜µ)Π := Π(L2

n(p) − ˜µ)Π : H2(R) ∩ X⊥

n (p) → ΠH−2(R).

(4.1.18)

spectra of Π(L2

The operator Π(L2

n(p) − ˜µ)Π is self adjoint, so its spectrum is real-valued. If the point
n(p) − ˜µ)Π is strictly positive for some values of ˜µ then we obtain (4.1.31).
n(p) − ˜µ)Π is strictly positive. By

It remains to show that the point spectra of Π(L2

Proposition 5.3.1 in [Kapitula and Promislow, 2013], number of the negative eigenvalues of

the constrained operator is given by the difference between the number of the negative

eigenvalues of the operator and the constrained matrix, D(˜µ),namely,

n((L2

n(p) − ˜µ)Π) = n(L2

n(p) − ˜µ) − n(D(˜µ)),

(4.1.19)

51

where n represents the count of the negative eigenvalues and D(˜µ) is given as

Dij(˜µ) := (cid:104)φ(cid:48)(z − pi), (L2

n(p) − ˜µ)−1φ(cid:48)(z − pj)(cid:105).

(4.1.20)

We first calculate the number of negative eigenvalues of the operator L2

n(p) − ˜µ by directly

examining the spectrum of the operator.

Since u1 = φh(z − p1) and φh is translation invariant, the linearized operator L about
φh and L1(p) are the same operators. Hence, we have that σ(L) = σ(L1(p)). On the other
hand, the essential spectrum of L1 is

σess(L1) = {−k2 − α− : k ∈ R}.

(4.1.21)

Further, σess(Ln(p)) = σess(L1(p)) by the classical Weyl essential spectrum theorem since
the operators have the same limiting states, limz→±∞un. (See Chapter 3 in [Kapitula and Promislow, 2013].)
On the other hand, to each point spectrum λk of L1(p), there are associated n eigenvalues
of Ln(p), {λk,j}n
|λk − λk,j| decays exponentially with growing pulse

j=1 such that max
j=1,...,n

separation.(See [Sandstede, 1998].)

Since Ln(p) is a self-adjoint operator, by the spectral mapping theorem σ(L2

n(p)) =
(σ(Ln(p)))2. Recall that in Remark 2.2.1 we present that the eigenvalues of Ln has an order
λ0 < 0 = λ1 < λ2 < . . . . By the choice of ˜µ > 0, the eigenvalues of L2
n(p) associated to
the eigenvalue of L1(p) at λ1 = 0 are shifted to the left by ˜µ(See Figure 4.1). Choosing
n(p) − ˜µ are
˜µ > 0 but less than the minimum of λ2
positive except n eigenvalues of Ln(p) associated to λ1 = 0, {λ1,j}n

j=1.

0 and λ2

2 we see that the eigenvalues of L2

We conclude that

n(L2

n(p) − ˜µ) = n.

52

(4.1.22)

(a) Spectrum of Ln(p)

(b) Spectrum of L2

n(p)

Figure 4.1: Figure (a) is a depiction for the spectrum of Ln(p). In the descending order
λ0 > 0 = λ1 > λ2 > ... (red disks) are the eigenvalues of L1(p) = L. Ln(p) has n associated
eigenvalues(black crosses) to each localized eigenvalue of L1(p) such that |λk − λk,j|j=1,...,n
decays exponentially with growing pulse separation. Figure (b) demonstrates the spectrum
for L2

n(p).

The next step is calculating the number of negative eigenvalues of the constrained matrix,
n(D(˜µ)). The eigenfunctions of L2

n(p)− ˜µ corresponding to the eigenvalues {λ2

(cid:96)=1 are in
h(z−pj) up to exponentially small terms. (See [Sandstede, 2001].)

the form ψ1,i =(cid:80)n

j=1 βijφ(cid:48)

1,i}n

Using the definition of D we have the identity

n(cid:88)

j=1

Dij(˜µ)βij = (cid:104)ψ1,i, (L2

n(p) − ˜µ)−1φ(cid:48)

h(z − pi)(cid:105).

(4.1.23)

Since inverse of a self-adjoint operator is also self-adjoint, transposing (L2

n(p) − ˜µ)−1 onto

(4.1.24)

ψ1,i provides

n(cid:88)

j=1

(cid:68) ψ1,i

λ1,i − ˜µ

(cid:69)

.

, φ(cid:48)

h(z − pi)

Dij(˜µ)βij =

53

σ(Ln)σess(Ln)λ2λ1=0λ0σ(L2n)σess(L2n)λ2λ1=0λ0222Then, inserting the formula of ψ1,i we obtain

j=1 βijφ(cid:48)

h(z − pj)

λ1,i − ˜µ

(cid:69)
h(z − pi)

, φ(cid:48)

(cid:68)(cid:80)n

n(cid:88)

j=1

Dij(˜µ)βij =

+ exp. small

terms

(4.1.25)

=

βii

λ1,i − ˜µ

+ exp. small

terms.

Let A be the matrix of the coefficients βij of eigenfunction ψ1,i which is comprised of the

vector of columns Bi = (βi1, βi2, . . . , βin)T . Then, by (4.1.25) we have

DBi =

1

λ1,i − ˜µ

Bi,

(4.1.26)

up to exponentially small terms. Hence, we conclude that
a value of ˜µ ∈ R for which

1

λ1,i − ˜µ

∈ σ(D) and there exists

min

i=1,...,n

λ2,i > ˜µ > max
i=1,··· ,n

λ1,i ∀ i = 1, . . . , n,

(4.1.27)

that guarantees

n(D(˜µ)) = n.

(4.1.28)

If we insert (4.1.22) and (4.1.28) in (4.1.19), we obtain the desired result as

n((L2

n(p) − ˜µ)Π) = 0,

(4.1.29)

which implies the L2 coercivity of L2

n(p), i.e. there exists a ˜µ > 0 independent of domain

size such that

(cid:104)L2

n(p)v, v(cid:105) ≥ ˜µ(cid:107)v(cid:107)2

L2 ,

(4.1.30)

54

for all v ∈ X⊥

n (p).

Corollary 4.1.5. There exists a ˜µ1 > 0, independent of the domain size, such that

(cid:104)L2

n(p)v, v(cid:105) ≥ ˜µ1 (cid:107)v(cid:107)2

H2 ,

(4.1.31)

for all v ∈ X⊥

n (p).

Proof. We already proved the L2-coercivity of the bilinear form induced by L2

n(p) in Lemma

4.1.4. We utilize this result to establish the H2-coercivity of the bilinear form induced by
L2
n(p). By the L2-coercivity of L2

n, there exists a constant ˜µ > 0 such that

(cid:104)Lnv,Lnv(cid:105) ≥ ˜µ(cid:107)v(cid:107)2

L2(R)

,

(4.1.32)

for all v ∈ Xn

⊥(p). Expanding the inner product, we fix θ ∈ (0, 1) and write

(cid:104)Lnv,Lnv(cid:105) = θ(cid:104)Lnv,Lnv(cid:105) + (1 − θ)(cid:104)Lnv,Lnv(cid:105),

∂2
z v

+ P

∂2
z v

v + Qv2dz + (1 − θ)˜µ(cid:107)v(cid:107)2

L2 ,

(4.1.33)

where P := −2W(cid:48)(cid:48)(un) and Q := (W (un))2. Applying the Holder’s inequality to the term

with P ,(cid:13)(cid:13)v∂2
z v(cid:13)(cid:13)L1

(cid:16)

(cid:17)

we obtain

(cid:17)2
2v(cid:13)(cid:13)L2

R

(cid:90)

≥ θ

(cid:16)
(cid:13)(cid:13)∂z
(cid:18)(cid:13)(cid:13)(cid:13)∂2
(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)∂2

≤ (cid:107)v(cid:107)L2

z v

z v

L2

(cid:104)Lnv,Lnv(cid:105) ≥ θ

≥ θ

− (cid:107)P(cid:107)L∞ (cid:107)v∂zv(cid:107)L1 − (cid:107)Q(cid:107)L∞ (cid:107)v(cid:107)2
L2 ,
L2
L2
− θ (cid:107)P(cid:107)L∞ (cid:107)∂zv(cid:107)L2 (cid:107)v(cid:107)L2 + ((1 − θ)˜µ − θ (cid:107)Q(cid:107)L∞)(cid:107)v(cid:107)2

+ (1 − θ)˜µ(cid:107)v(cid:107)2

L2 .

(cid:19)

(4.1.34)

55

We apply Young’s inequality to the second term on the second line of (4.1.34),

(cid:19)

L2 + ((1 − θ)˜µ − θ (cid:107)Q(cid:107)L∞)(cid:107)v(cid:107)2
L2 ,
(cid:107)P(cid:107)2
(cid:17)(cid:111)

(cid:107)v(cid:107)2

L∞

L2 .

(4.1.35)

independent of domain size we

.

(4.1.36)

(cid:13)(cid:13)(cid:13)2

(cid:13)(cid:13)(cid:13)∂2
(cid:18)
L∞ (cid:107)v(cid:107)2
(1 − θ)˜µ − θ (cid:107)Q(cid:107)L∞ − θ
2

− θ
2

(cid:107)P(cid:107)2

z v

L2

− θ
2

+

(cid:104)Lnv,Lnv(cid:105) ≥ θ
θ
2

=

L2

z v

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)∂2
(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)∂2
(cid:16)
(cid:110) θ

z v

L2

,

2

Choosing θ∗ = min

(1 − θ)˜µ − θ (cid:107)Q(cid:107)L∞ − θ

2 (cid:107)P(cid:107)2
L∞

obtain

(cid:104)Lnv,Lnv(cid:105) ≥ θ∗

(cid:19)

+ (cid:107)v(cid:107)2
L2

(cid:18)(cid:13)(cid:13)(cid:13)∂2

z v

(cid:13)(cid:13)(cid:13)2

L2

= θ∗ (cid:107)v(cid:107)2

H2 .

Proof of Theorem 4.1.2. We show the H2-coercivity of the second variation of I about

n-pulse ansatz using bilinear form given in (4.1.11).

(cid:68)Lv, v

(cid:69)

=

(cid:68)(cid:18)
+ −(cid:16)−δ
n(cid:88)
(cid:69) − δ¯λ
(cid:68)Ln(W(cid:48)(cid:48)(cid:48)(un)B2,nv), v
n(cid:88)

(cid:17)
nv − δ¯λLn(W(cid:48)(cid:48)(cid:48)(un)B2,nv) − δ¯λW(cid:48)(cid:48)(cid:48)(un)B2,nLnv
L2
(cid:17)
(cid:69)
W(cid:48)(cid:48)(cid:48)(un)v, v
(cid:69) − δ¯λ
(cid:68)
W(cid:48)(cid:48)(cid:48)(un)B2,nLnv, v
(cid:69)
(cid:69) − −δ¯λ
(cid:68)
W(cid:48)(cid:48)(cid:48)(un)B1,nv, v

W(cid:48)(cid:48)(φh(z − pi)) + δ¯λB1,n

nv, v
W(cid:48)(cid:48)(cid:48)(un)

W(cid:48)(cid:48)(φh(z − pi))v, v

− δ

i=1

+

(cid:104)Lv, v(cid:105) ≥(cid:68)L2
(cid:68)

(4.1.37)

+ O(δ2 (cid:107)v(cid:107)2

H2)

(cid:69)

+

+

(4.1.38)

i=1

+ O(δ2 (cid:107)v(cid:107)2

H2).

From Corollary 4.1.5, there exists a ˜µ1 > 0 independent of ε such that (cid:104)L2

nv, v(cid:105) ≥ ˜µ1 (cid:107)v(cid:107)2
H2

56

and hence

(cid:68)Ln(W(cid:48)(cid:48)(cid:48)(un)B2,nv), v
H2 − δ¯λ
n(cid:88)
W(cid:48)(cid:48)(cid:48)(un)

W(cid:48)(cid:48)(φh(z − pi))v, v

(cid:69) − δ¯λ
(cid:68)
W(cid:48)(cid:48)(cid:48)(un)B2,nLnv, v
(cid:69)
(cid:69) − δ¯λ
(cid:68)
W(cid:48)(cid:48)(cid:48)(un)B1,nv, v

+

(cid:69)

+

(4.1.39)

(cid:104)Lv, v(cid:105) ≥ ˜µ1 (cid:107)v(cid:107)2
(cid:68)

− δ

i=1

+ O(δ2 (cid:107)v(cid:107)2

H2).

By the smoothness of the functions W over bounded functions and the smoothness of B1,n
and B2,n and by the definition of Ln we get the following estimates for the inner products
in (4.1.39). Since Ln is self-adjoint, transposing that onto v we have

(cid:68)Ln(W(cid:48)(cid:48)(cid:48)(un)B2,nv), v

(cid:69)

=

W(cid:48)(cid:48)(cid:48)(un)B2,nv,Lnv

(cid:68)
(cid:69)
=(cid:13)(cid:13)W(cid:48)(cid:48)(cid:48)(un)B2,nvLnv(cid:13)(cid:13)L1
(cid:13)(cid:13)L∞(cid:13)(cid:13)vLnv(cid:13)(cid:13)L1
≤(cid:13)(cid:13)W(cid:48)(cid:48)(cid:48)(un)(cid:13)(cid:13)L∞(cid:13)(cid:13)B2,n

(4.1.40)

(4.1.41)

Then, we apply the Holder’s inequality to the term with L1 norm

(cid:68)Ln(W(cid:48)(cid:48)(cid:48)(un)B2,nv), v

(cid:69) ≤(cid:13)(cid:13)W(cid:48)(cid:48)(cid:48)(un)(cid:13)(cid:13)L∞(cid:13)(cid:13)B2,n

(cid:13)(cid:13)L∞(cid:13)(cid:13)Lnv(cid:13)(cid:13)L2 (cid:107)v(cid:107)L2 .

Here we use the fact that there exists a constant c > 0 such that(cid:13)(cid:13)Lnf(cid:13)(cid:13)L2 ≤ c(cid:107)f(cid:107)H2 for

all f ∈ H2(R) and get

(cid:68)Ln(W(cid:48)(cid:48)(cid:48)(un)B2,nv), v

(cid:69) ≤ c(cid:13)(cid:13)W(cid:48)(cid:48)(cid:48)(un)(cid:13)(cid:13)L∞(cid:13)(cid:13)B2,n

(cid:13)(cid:13)L∞ (cid:107)v(cid:107)H2 (cid:107)v(cid:107)L2

≤ θ1 (cid:107)v(cid:107)2

H2 ,

(4.1.42)

for some θ1 > 0. With a similar calculation, we get an upper bound for the second inner

57

product,(cid:68)

(cid:69)

W(cid:48)(cid:48)(cid:48)(un)B2,nLnv, v

=

(cid:13)(cid:13)(cid:13)W(cid:48)(cid:48)(cid:48)(un)B2,nLnv2(cid:13)(cid:13)(cid:13)L1
(cid:13)(cid:13)(cid:13)Lnv2(cid:13)(cid:13)(cid:13)L1
(cid:13)(cid:13)L∞
≤(cid:13)(cid:13)W(cid:48)(cid:48)(cid:48)(un)(cid:13)(cid:13)L∞(cid:13)(cid:13)B2,n
(cid:13)(cid:13)L∞(cid:13)(cid:13)Lnv(cid:13)(cid:13)L2 (cid:107)v(cid:107)L2
≤(cid:13)(cid:13)W(cid:48)(cid:48)(cid:48)(un)(cid:13)(cid:13)L∞(cid:13)(cid:13)B2,n
(cid:13)(cid:13)L∞(cid:13)(cid:13)Lnv(cid:13)(cid:13)2
≤(cid:13)(cid:13)W(cid:48)(cid:48)(cid:48)(un)(cid:13)(cid:13)L∞(cid:13)(cid:13)B2,n

H2

≤ θ2 (cid:107)v(cid:107)2

H2 ,

(4.1.43)

for some θ2 > 0. For the remaining two inner products, there exist constants θ3 > 0 and

θ4 > 0 such that

and

(cid:68)

W(cid:48)(cid:48)(cid:48)(un)
(cid:68)

n(cid:88)

i=1

W(cid:48)(cid:48)(φh(z − pi)v, v

(cid:69) ≤ θ3 (cid:107)v(cid:107)2

H2 ,

W(cid:48)(cid:48)(cid:48)(un)B1,nv, v

(cid:69) ≤ θ4 (cid:107)v(cid:107)2

H2 .

(4.1.44)

(4.1.45)

Inserting the estimates (4.1.42), (4.1.43), (4.1.44) and (4.1.45) in (4.1.39) provides us

(cid:104)Lv, v(cid:105) ≥ ˜µ1 (cid:107)v(cid:107)2

H2 − δ¯λθ1 (cid:107)v(cid:107)2

− δθ3 (cid:107)v(cid:107)2

H2 − δ¯λθ4 (cid:107)v(cid:107)2

H2 − δ¯λθ2 (cid:107)v(cid:107)2

H2 +
H2 + O(δ2 (cid:107)v(cid:107)2

H2).

(4.1.46)

Choosing the ˜µ2 = max{¯λθ1, ¯λθ2, θ3, ¯λθ4}, we obtain the H2-coercivity of

δ2I
δu2 (Φn) as

(cid:104)Lv, v(cid:105) ≥ ˜µ1 (cid:107)v(cid:107)2

H2 − δ ˜µ2 (cid:107)v(cid:107)2

H2 .

(4.1.47)

We also show the H2-coercivity of the L2

n(p) restricted to the bounded domain [0,

L2
ε ]

58

that is an essential step in the verification of the assumption (H3) in Section 4.2.

Corollary 4.1.6. Let the operator Ln,b(p) acting on H2(cid:16)(cid:104)

(cid:105)(cid:17)

0,

L2
ε

be the restriction of the

Ln(p) given in (2.2.4) to the large bounded domain [0,
by L2

n,b(p) is coercive, i.e, there exists a µ∗ > 0 independent of ε such that

L2
ε ]. Then, the bilinear form induced

(cid:104)L2

n,b(p)v, v(cid:105) ≥ µ∗(cid:107)v(cid:107)

(cid:18)(cid:20)

H2

0,

L2
ε

(cid:21)(cid:19),

(4.1.48)

for all v ∈ Y (cid:48)

n(p).

Proof. Let ˜v ∈ Y (cid:48)

n(p) have a compact support. Let v ∈ H2(R) be the extension of ˜v to R

so that we have

(cid:104)L2

n,b(p)v, v(cid:105) = (cid:104)L2

n(p)v, v(cid:105),

(4.1.49)

for all v ∈ X⊥

n (p). By Corollary 4.1.5, there exists a ˜µ1 such that

(cid:104)L2

n,b(p)v, v(cid:105) = (cid:104)L2

n(p)v, v(cid:105) ≥ ˜µ1(cid:107)v(cid:107)H2(R),

n (p). Since H2(cid:16)(cid:104)

0,

L2
ε

(cid:105)(cid:17)

with compact support is dense in H2(cid:16)(cid:104)

for all v ∈ X⊥

(4.1.50)

(cid:105)(cid:17)

0,

L2
ε

the

result follows.

4.1.3 H 2-coercivity of L2

n

For the stability of the periodic multi-pulse solutions, Φn, given in (2.2.37) of the free energy

I, the H2-coercivity of the second variation of I about Φn at leading order is established
utilizing the H2-coercivity of the operator L2

n. See Appendix for the derivation of second

variation of I about Φn at leading order.

59

Theorem 4.1.7. Consider the operator Ln given in (2.2.27) on H2(cid:16)(cid:104)

(cid:105)(cid:17)

0,

L2
ε

which is the

linearization of (1.2.6) about periodic multi-pulse solutions, φn, at the leading order. Then,
n is coercive, i.e, there exists a µ∗ > 0 independent

the bilinear form induced by

δ2I
δu2 (un) = L2

of ε such that

for all v ∈ X⊥

n (p).

nv, v(cid:105) ≥ µ∗(cid:107)v(cid:107)H2(R),
(cid:104)L2

(4.1.51)

Lemma 4.1.8. There exist smooth functions (p, η) : H2 → RN × H2 satisfying p(0) = p∗
such that for all (cid:107)v(cid:107)H2 = O(1) sufficiently small

Φn + v = Φn(z, p, ¯λ) + η(v),

(4.1.52)

where η(v) ∈ X⊥

n (p). Further, there exists α > 0 constant such that

(cid:107)p(v) − p∗(cid:107)L2 ≤ α(cid:107)v(cid:107)H2,

(4.1.53)

where v ∈ X⊥

n (p).

Proof. Introduce F = (F1, ..., Fn)t where

Fi(v, p) := (cid:104)v + Φn(p) − Φn(p), φ(cid:48)

n(z − pi)(cid:105),

(4.1.54)

for i = 1, ..., n and (cid:104)·,·(cid:105) is L2 inner product. Fi = 0 for each i since η(v) = v+Φn(¯p)−Φn(p) ∈
X⊥
n (p). Indeed, Fi for each i attains its minima for some values of v on the compact set of
p ∈ P ⊂ Rn. Assume that F attains its minima at p(v0) and so, p = p(v0) is one solution

to F (v, p) = 0 for v = v0. We apply the Implicit Function Theorem to F (v, p). The ij th

60

entry of the gradient ∇pF is

(cid:18)

∇pF

(cid:12)(cid:12)(cid:12)(v0,p(v0)

(cid:19)

=

ij

=

h(z − pi)(cid:105)(cid:17)(cid:12)(cid:12)(cid:12)(v0,p(v0))

,

i = j,

i (cid:54)= j,




(cid:16)−(cid:104)∂Φn(p)

, φ(cid:48)

∂pi

h(z − pi)(cid:105)+
+ (cid:104)v + Φn − Φn(p),−φ(cid:48)(cid:48)

, φ(cid:48)
h(z − pi(v(0)), φ(cid:48)

h(z − pi)(cid:105)(cid:12)(cid:12)(cid:12)(v0,p(v0))
h(z − pi(v0))(cid:105)(cid:17)

+ (cid:104)h,−φ(cid:48)(cid:48)

(cid:104) ∂Φn(p)
∂pj

(cid:16)−(cid:104)φ(cid:48)

,

h(z − pi(v0))(cid:105)+

,

0,

i = j,

i (cid:54)= j,

(4.1.55)

where h := v0 + Φn − Φn(p(v0)) is small with (cid:107)h(cid:107)H2 = O(δ). By a proper choice of far
apart pulse locations p ∈ P, φ(cid:48)(z − pi) satisfy

 κ for

0 for

(φ(cid:48)
h(z − pi), φ(cid:48)

h(z − pj)2 =

i = j,

i (cid:54)= j,

(4.1.56)

for constant κ > 0. From this relation, we write the gradient as

(cid:12)(cid:12)(cid:12)(v0,p(v0)

∇pF

= −κI + O(δ).

(4.1.57)

We see that ∇pF

(cid:12)(cid:12)(cid:12)(v0,p(v0)

has non-zero determinant, and from this the Implicit Function

Theorem implies that there exists a neighborhood (v0, p(v0)) and a unique function p(v)

such that F (p(v), v) = 0. Further, by the Implicit Function Theorem p is smooth since Fi

61

is smooth. Thus, the Taylor expansion of p about v = 0 yields

p(v) = p(0) + (cid:104) δp
δv

(0), v(cid:105)H2 + O((cid:107)v(cid:107)2

H2).

(4.1.58)

From this, we obtain

|p(v) − p(0)| ≤

(cid:13)(cid:13)(cid:13)(cid:13) δp

δv

(cid:13)(cid:13)(cid:13)(cid:13)H−2

(0)

for a constant θ > 0.

(cid:107)v(cid:107)2

H2 + c((cid:107)v(cid:107)2

H2) ≤ θ(cid:107)v(cid:107)2

H2,

(4.1.59)

Proof of Theorem 4.1.7. We prove the H2-coercivity of L2
L2
n(p). Expanding the bilinear form induced by L2
by L2

n(p) and utilizing H2 coercivity of the bilinear form induced by L2

n(p) from Corollary

n(p) using H2-coercivity of

n(p) in terms of the bilinear form induced

4.1.5, we obtain a lower bound for the first term in the expansion,

(cid:104)L2

n(p)v, v(cid:105) = (cid:104)L2

n(p)v, v(cid:105) + (cid:104)(L2
H2 + (cid:104)(L2

n(p) − L2
n(p) − L2

n(p))v, v(cid:105),
n(p))v, v(cid:105),

≥ ˜µ1 (cid:107)v(cid:107)2

(4.1.60)

for all v ∈ X⊥

n (p) with (cid:107)v(cid:107)H2 (cid:28) 1. Hence, it suffices to show that the remaining term in

the expansion (4.1.60) is small. We attack this term splitting it into

(cid:104)(L2

n(p) − L2

n(p))v, v(cid:105) = (cid:104)(L2

n(p) − L2

n(p∗))v, v(cid:105) + (cid:104)(L2

n(p∗) − L2

n(p))v, v(cid:105).

(4.1.61)

62

Let u∗

n := un(z, p∗).
(cid:17)

n(p∗) − L2

n(p)

(cid:16)L2

Define

v,

(cid:17)2
z − W(cid:48)(cid:48)(un)
∂2
n) − 2∂zv∂z(W(cid:48)(cid:48)(u∗
z vW(cid:48)(cid:48)(u∗

n)(cid:1)2
n)) +(cid:0)W(cid:48)(cid:48)(u∗
z vW(cid:48)(cid:48)(un) − 2∂zv∂z(W(cid:48)(cid:48)(un)) +(cid:0)W(cid:48)(cid:48)(un)(cid:1)2
(cid:0)W(cid:48)(cid:48)(un)(cid:1) ,

(4.1.62)

(cid:17)
(cid:17)

v

v

,

.

v =

(cid:16)
(cid:16)
−(cid:16)

=

z

z

z − W(cid:48)(cid:48)(u∗
∂2
n)
z v − ∂2
∂4
z v − ∂2
∂4

(cid:17)2
v −(cid:16)
n)(cid:1) v − 2∂2
(cid:0)W(cid:48)(cid:48)(u∗
(cid:0)W(cid:48)(cid:48)(un)(cid:1) v − 2∂2
A0(z, p) :=(cid:0)W(cid:48)(cid:48)(un)(cid:1)2 − ∂2
(cid:0)W(cid:48)(cid:48)(un)(cid:1) ,

A1(z, p) := −2∂z
A2(z, p) := −2W(cid:48)(cid:48)(un),

z

(4.1.63)

(4.1.64)

which are smooth functions in z. Then,

(cid:16)L2

n(p∗) − L2

n(p)

(cid:17)

v = (A0(z, p∗) − A0(z, p)) v,

+ (A1(z, p∗)∂zv − A1(z, p)) ∂zv,
(cid:17)
(cid:16)
z v − A2(z, p)

A2(z, p∗)∂2

+

∂2
z v,

and so, by the Mean Value Theorem for several variable functions, there exist ξ0, ξ1, ξ2 ∈

[p(0), p(v)] so that

(cid:10)(cid:16)L2

n(p∗) − L2

n(p)

(cid:17)

v, v(cid:11) ≤

(cid:90) L2

ε

0

(A0(z, p) − A0(z, p∗)) v2 + (A1(z, p) − A1(z, p∗)) v∂zv

+ (A2(z, p) − A0(z, p)) v∂2
(cid:90) L2

z vdz,

ε

|∇pA0(ξ0)||p(v) − p∗|v2 + |∇pA1(ξ1)||p(v) − p∗|v∂zv

≤

0

+ |∇pA2(ξ2)||p(v) − p∗|v∂2

z vdz,

63

ε

(cid:90) L2
≤(cid:13)(cid:13)∇pA0(ξ0)(cid:13)(cid:13)L∞
|p(v) − p∗|v2dz+
(cid:90) L2
+(cid:13)(cid:13)∇pA1(ξ1)(cid:13)(cid:13)L∞
(cid:90) L2
+(cid:13)(cid:13)∇pA2(ξ2)(cid:13)(cid:13)L∞

|p(v) − p∗|v∂2

|p(v) − p∗|v2dz+

0

0

ε

ε

z vdz.

0

Utilizing the bound for |p(v) − p∗| in Lemma 4.1.8,

(cid:10)(cid:16)L2

n(p) − L2

(cid:17)
n(p∗)

v, v(cid:11) ≤ α(cid:107)v(cid:107)H2

(cid:90) L2

ε

0

v2dz

(cid:18)(cid:13)(cid:13)∇pA0(ξ0)(cid:13)(cid:13)L∞
(cid:90) L2
(cid:90) L2

+(cid:13)(cid:13)∇pA1(ξ1)(cid:13)(cid:13)L∞
+(cid:13)(cid:13)∇pA2(ξ2)(cid:13)(cid:13)L∞

0

ε

ε

0

v∂zvdz+

(cid:19)

v∂2

z vdz

.

{(cid:13)(cid:13)∇pAj(ξj)(cid:13)(cid:13)L∞}.
(cid:17)
v, v(cid:11) ≤ Cα(cid:107)v(cid:107)H2
n(p∗)

Now let C = max
1≤j≤3

(cid:10)(cid:16)L2

n(p) − L2

(4.1.65)

(4.1.66)

On the other hand, a similar calculation gives a bound for the second term in (4.1.60)

(cid:104)(L2

n(p) − L2

n(p∗))v, v(cid:105) =

≤

(cid:16)

(cid:90) L2
(cid:90) L2

0

ε

ε

0

(cid:18)

(cid:107)v(cid:107)2

H2 + (cid:107)∂zv(cid:107)L2 (cid:107)v(cid:107)L2 + (cid:107)∂zv(cid:107)L2 (cid:107)v(cid:107)L2

(cid:19)

,

≤ C1(cid:107)v(cid:107)3

H2.

(4.1.67)

(cid:17)2 −(cid:16)

(cid:17)2

z − W(cid:48)(cid:48) (φn)
∂2

(

z − W(cid:48)(cid:48) (u∗
∂2
n)

)v2dz,

(A0(z) − A0(z, p∗)) v2 + (A1(z) − A1(z, p∗)) v∂zv,

+ (A2(z) − A0(z, p∗)) v∂2
≤ C2 (cid:107)v(cid:107)2

H2 .

z vdz

(4.1.68)

64

Hence, inserting (4.1.67) and (4.1.68) in (4.1.61),

(cid:104)(L2

n(p) − L2

n(p))v, v(cid:105) ≥ −C1 (cid:107)v(cid:107)3

H2 − C2 (cid:107)v(cid:107)2

H2 ,

and (4.1.69) in (4.1.60) we obtain the H2-coercivity

(cid:104)L2

n(p)v, v(cid:105) ≥ ˜µ1 (cid:107)v(cid:107)2
≥ α(cid:107)v(cid:107)2

H2 + (cid:104)(L2
H2 − C1 (cid:107)v(cid:107)3

n(p) − L2

n(p))v, v(cid:105),

H2 .

(4.1.69)

(4.1.70)

4.2 Modulational Stability of n-Pulses

We establish the modulational stability for n-pulse ansatz, Φn, with respect to the Π0

gradient flow of the inner scaling of the FCH energy given (1.2.3) with an application

of Theorem 2.1 in [Promislow, 2002] where modulational stability of manifolds of quasi-

stationary solutions to dispersive equations is established.

Introduce the Π0-gradient flow of the inner scaling of the FCH free energy, I, given (1.2.3)

ut = −Π0

δI
δu

(u),

(4.2.1)

where δI

δu is the first variational derivative of I with respect to L2 inner product and Π0 is

the mass preserving L2 projection

Π0f := f − ε
L2

65

(cid:90) L2

ε

0

f (z)dz.

(4.2.2)

Consider the family of multi-pulse critical points {Φn : n ∈ N} of the free energy I and

the n-dimensional manifold Mn = {Φn(p, ¯λ)|p ∈ P} where P ⊂ Rn defined in (4.1.4).

For the modulational stability of the n-pulses {Φn : n ∈ N}, we apply renormalization

techniques from [Promislow, 2002]. We are interested in the evolution of the solutions which
lie in a neighborhood of the manifold Mn which consists of the n-pulse solutions Φn(p, ¯λ)
for p ∈ P. Introduce

u(z, t) = Φn(z, p, ¯λ) + w(z).

(4.2.3)

We reduce the dynamics of (4.2.1) near the manifold Mn to a weakly non-linear flow which

is predominantly controlled by the terms that are linear in their deviation, w, of U from
Mn, and non-linear in the pulse evolution p = p(t) about Φn(p, ¯λ). Taylor expanding the
flow (4.2.1) about Φn(p, ¯λ) we obtain

− Π0

δI
δu

(u(z, t)) = −Π0

(Φn(p, ¯λ))

− Π0L2

n(p)w + N (w),

(4.2.4)

(cid:18) δI

δu

(cid:19)

where Ln(p) given (2.2.4) is the linearization about un and N represents the non-linear

terms. The linearization about un will be weakly time dependent through the slow evolution

of the pulse positions, p, and the background state ¯λ. The n + 1 parameters form the

coordinates of the slow n-pulse manifold, one of which is determined by the mass constraint.

In Theorem 2.1 of [Promislow, 2002], there are some assumptions on the linearized

operator and the manifold of the steady-state solutions to the gradient flow. Here we present
those assumptions adapting to the linearized operator −Π0L2
n(p) and the manifold Mn but

defer the verification of each assumption to the proof of Theorem 4.2.1.

(H0) The manifold Mn is quasi-steady, i.e, for δ > 0, the scaling of ¯λ, there exists M > 0

66

such that

(cid:13)(cid:13)(cid:13)(cid:13)−Π0

δI
δu

(cid:13)(cid:13)(cid:13)(cid:13)H2

(Φn(p, ¯λ))

≤ M δ.

(4.2.5)

(H1) The spectrum of each operator −Π0L2

n(p) consists of a stable part σs ⊂ {λ|λ ≤

−ks} for some ks > 0 and a slow part σ0 ⊂ {λ||λ| ≤ c0e− k0

ε } for some c0, k0 > 0.

(H2) Each operator −Π0L2

n(p) generates a C0 semigroup Sp which satisfies

(cid:13)(cid:13)Sp(t)u(cid:13)(cid:13)H2 ≤ M e−kst (cid:107)u(cid:107)H2 ,

n (p)∩ H2(cid:16)(cid:104)

(cid:105)(cid:17)

0,

L2
ε

, where X⊥

n (p) is perpendicular to the tangent

(4.2.6)

for all t ≥ 0, u ∈ Y (cid:48)

n := X⊥
plane Xn(p) of Φn(p, ¯λ).

Let Yp represent the slow space of the linearized operator −Π0L2

n(p), the n+1 dimensional

space associated with small eigenvalues of −Π0L2

n(p).

Recall that Xn(p) = span{ ∂Φn(p)
∂pi

(cid:110){φ(cid:48)
plane to the manifold Mn. Introduce Xn+1(p) = span
(cid:18)(cid:20)

: p ∈ P ⊂ Rn} = span{φ(cid:48)

h(z − pi)}n
h(z − pi)}n
(cid:21)(cid:19)

i=1 ∪ {B2,n}(cid:111)

and

i=1 is the tangent

Y (cid:48)
n+1(p) := X⊥

n+1(p) ∩ H2

0,

L2
ε

,

(4.2.7)

where X⊥

n+1(p) is the orthogonal space to Xn+1(p).

(H3) Yp is well-approximated by Xn+1(p).

In [Promislow, 2002], this assumption is

utilized to establish the coercivity of the linearized operator. Instead, here we will establish
the H2-coercivity of Π0L2
of L2

n+1(p) which follows from the H2-coercivity

n(p). Recall that H2-coercivity of L2

n(p) over the space Y (cid:48)

n,b(p) over Y (cid:48)

n,b(p) over Y (cid:48)

n(p) was established in

Corollary 4.1.6 using Corollary 4.1.5.

We assume that the adjoint of the elements in Yp and Xn+1(p) satisfy the following

67

normalization condition

1

0

(cid:104)ζi, ζ

†
j(cid:105) =

for

i = j,

for

i (cid:54)= j.

With this condition, the adjoint of Xn+1(p) is defined as

†
n+1(p):=Xn(p)∪span{1}.

X

(H4) The normalized adjoint eigenvectors {ψ

†
1, ..., ψ

†
n+1} of the space Yp satisfy

(cid:16)(cid:13)(cid:13)(cid:13)ψ

†
i (p)

(cid:13)(cid:13)(cid:13)H2

max
i=1,...,n
p∈P

for some M .

(cid:13)(cid:13)(cid:13)∇2

+

†
i (p)

pψ

(cid:17) ≤ M,

(cid:13)(cid:13)(cid:13)H2

(4.2.8)

(4.2.9)

(4.2.10)

Theorem 4.2.1. Fix a pulse separation value (cid:96) = O(ε−1) > 0 in P ⊂ Rn. Then, there exists
a manifold Mn = {Φn(p, ¯λ)|p ∈ P} satisfying the hypothesis (H0)-(H4) for some constants
M and k and there exist ε0, M0 for ε ∈ [0, ε0] such that for all initial data u(z, t0) = u0(z)
within ε2-neighborhood in H2-norm of Mn whose mass lies within δ-neighborhood of the
mass of Φn(p, ¯λ), the solution u of (4.2.1) can be decomposed as

u(z) = Φn(z, p(t), ¯λ) + w(z, t),

(4.2.11)

where the deviation w ∈ Y (cid:48)

n+1(p(t)) satisfies

(cid:107)w(·, t)(cid:107)H2 ≤ M0(ε2e−ks(t−t0) + δ)

f or

t ≥ t0.

(4.2.12)

The pulse locations p(t) = (p1, . . . , pn)t may be chosen to lie on a smooth curve in P. After

68

an initial transient T i ∼ 1| ln(ε)|, that is, for t > t0 + T i, the evolution of the pulse locations

is governed to leading order by the closed system

(cid:68) − Π0

p(cid:48)
i =

δI
δu

(Φn(p, ¯λ)),

1(cid:13)(cid:13)φ(cid:48)
(cid:13)(cid:13)2

h

L2

φ(cid:48)
h(z − pi)

(cid:69)

+ O(δ2)

f or

t ≥ t0 + T i,

(4.2.13)

for i = 1, . . . , n.

Proof. Here we verify that the Mn and the linearized operator −Π0L2

n(p) satisfy the

hypothesis (H0)-(H4) and a direct application of Theorem 2.1 from [Promislow, 2002] provides

the result.

(H0) We prove that the manifold Mn is quasi-steady, namely,

(cid:13)(cid:13)(cid:13)(cid:13)−Π0

δI
δu

(cid:13)(cid:13)(cid:13)(cid:13)H2

(Φn(p, ¯λ))

≤ M δ,

for some M > 0. Recall that Φn(p, ¯λ) = un + δ¯λB2,n where un =(cid:80)n

j=1 φh

with φh = φh − b−. The Taylor series expansion of δI

(4.2.14)

(cid:0)z − pj

(cid:1) + b−

δI
δu

(Φn(p, ¯λ)) =

=

=

(cid:16)
(cid:16)
(cid:16)

δu about un provides

(cid:17)(cid:16)
(cid:17)
z Φn(p) − W(cid:48)(Φn(p))
z − W(cid:48)(cid:48)(Φn(p))
(cid:17)(cid:16)
∂2
∂2
z − W(cid:48)(cid:48)(un + δ¯λB2,n
(cid:17)(cid:16)
∂2
z − W(cid:48)(cid:48)(un) − δ¯λB2,nW(cid:48)(cid:48)(cid:48)(un)
z (B2,n)+
z un + δ¯λ∂2
∂2
∂2
δ2¯λ2(B2,n)2W(cid:48)(cid:48)(cid:48)(un)
− W(cid:48)(un) − δ¯λB2,nW(cid:48)(cid:48)(un)) − 1
(cid:105)

(cid:17)
z (un + δ¯λB2,n) − W(cid:48)(un + δ¯λB2,n)
∂2
(cid:17)

2

.

(cid:104) pi−1+pi

Since the tail-tail interaction of the adjacent pulses dominates the value of un, on each
we can write un = φi−1 + φi + φi+1 where φi = φh(z − pi) and
δu(Φn(p, ¯λ)) about φi we

window
φi = φi − b−. Letting φ∆i := φi−1 + φi+1 and Taylor expanding δI

pi+pi+1

2

2

,

(4.2.15)

69

obtain

δI
δu

(Φn(p, ¯λ)) =

i=1

∂2

(cid:1)(cid:17)(cid:16)

(cid:16)
(cid:17)(cid:16)
(cid:17)
z − W(cid:48)(cid:48)(Φn(p, ¯λ))
z Φn(p, ¯λ) − W(cid:48)(Φn(p, ¯λ))
(cid:16)
n(cid:88)
z − W(cid:48)(cid:48)(cid:0)φi + φ∆i + δλB2,n
∂2
∂2
(cid:17)
− W(cid:48)(cid:16)
z − W(cid:48)(cid:48)(φi) − W(cid:48)(cid:48)(cid:48)(φi)(cid:0)φ∆i + δ¯λB2,n)(cid:1)(cid:17)(cid:16)
(cid:16)
n(cid:88)
zB2,n − W(cid:48)(φi) − W(cid:48)(cid:48)(φi)(cid:0)φ∆i + δ¯λB2,n

φi + φ∆i + δλB2,n

+ δ¯λ∂2

∂2

i=1

=

=

z (φi + φ∆i + δλB2,n)
∂2

z φi + ∂2
∂2

z φ∆i+

(cid:1)(cid:17)

.

(4.2.16)

Letting Li = ∂2

written as

δI
δu

(Φn(p, ¯λ)) =

i = W(cid:48)(φi), δI

z − W(cid:48)(cid:48)(φi) and taking into account that φ(cid:48)(cid:48)
(cid:16)Li − W(cid:48)(cid:48)(cid:48)(φi)
(cid:16)
n(cid:88)
(cid:16)Li − W(cid:48)(cid:48)(cid:48)(φi)φ∆i
n(cid:88)

(cid:17)Liφ∆i + δ¯λ

φ∆i + δ¯λB2,n

i=1

=

(cid:17)(cid:17)(cid:16)Liφ∆i + δ¯λLiB2,n
(cid:16)Li − W(cid:48)(cid:48)(cid:48)(φi)φ∆i

(cid:17)
(cid:17)LiB2,n+

δu(Φn(p, ¯λ)) can be

(4.2.17)

i=1

− δ¯λW(cid:48)(cid:48)(cid:48)(φi)B2,nLiφ∆i + O(δ2).

Applying −Π0 onto (4.2.17) and recalling the definition of B2,n given in (2.2.12) we rearrange

70

the terms

−Π0

δI
δu

(Φn(p, ¯λ)) = − n(cid:88)

Π0

i=1

= − n(cid:88)

i=1

Π0

(cid:17)LiB2,n+

(cid:18)(cid:16)Li − W(cid:48)(cid:48)(cid:48)(φi)φ∆i
(cid:16)L2

(cid:17)Liφ∆i + δ¯λ
(cid:16)Li − W(cid:48)(cid:48)(cid:48)(φi)φ∆i
(cid:19)
− δ¯λW(cid:48)(cid:48)(cid:48)(φi)B2,nLiφ∆i + O(δ2)
(cid:19)
− δ¯λW(cid:48)(cid:48)(cid:48)(φi)B2,nLiφ∆i + O(δ2)

i φ∆i − W(cid:48)(cid:48)(cid:48)(φi)φ∆iLiφ∆i + δ¯λ − δ¯λW(cid:48)(cid:48)(cid:48)(φi)φ∆iB1,n+

.

Since Π0 onto any constant is 0, Π0δ¯λ = 0 and hence,

−Π0

δI
δu

(Φn(p, ¯λ)) =

n(cid:88)

(cid:16)−Π0L2

i=1

+ δ¯λΠ0

(cid:0)W(cid:48)(cid:48)(cid:48)(φi)φ∆iLiφ∆i
(cid:1)(cid:17)
(cid:0)W(cid:48)(cid:48)(cid:48)(φi)B2,nLiφ∆i

i φ∆i + Π0

+ O(δ2).

(4.2.18)

(cid:1) + δ¯λΠ0

(cid:0)W(cid:48)(cid:48)(cid:48)(φi)φ∆iB1,n

(cid:1) +

Then, we calculate the H2-norm of (4.3.5). By the triangle inequality we have

(cid:13)(cid:13)(cid:13)(cid:13)−Π0

δI
δu

(Φn(p, ¯λ))

(cid:13)(cid:13)(cid:13)(cid:13)H2

i φ∆i

(cid:18)(cid:13)(cid:13)(cid:13)−Π0L2
≤ n(cid:88)
+ δ¯λ(cid:13)(cid:13)Π0
+ δ¯λ(cid:13)(cid:13)Π0

(cid:13)(cid:13)(cid:13)H2
+(cid:13)(cid:13)Π0
(cid:0)W(cid:48)(cid:48)(cid:48)(φi)φ∆iLiφ∆i
(cid:1)(cid:13)(cid:13)H2 +
(cid:0)W(cid:48)(cid:48)(cid:48)(φi)φ∆iB1,n
(cid:1)(cid:13)(cid:13)H2
(cid:0)W(cid:48)(cid:48)(cid:48)(φi)B2,nLiφ∆i

+ O(δ2).

(cid:19)

i=1

(4.2.19)

(cid:1)(cid:13)(cid:13)H2 +

(4.2.20)

Since Π0 is a H2-orthogonal projection, for any function u ∈ H2 we have the estimate

(cid:107)Π0u(cid:107)H2 ≤ (cid:107)u(cid:107)H2 ,

(4.2.21)

71

(cid:13)(cid:13)H2 +

(4.2.22)

i=1

i φ∆i

(cid:13)(cid:13)(cid:13)H2

+(cid:13)(cid:13)W(cid:48)(cid:48)(cid:48)(φi)φ∆iLiφ∆i
(cid:19)

(cid:18)(cid:13)(cid:13)(cid:13)L2
≤ n(cid:88)
+ δ¯λ(cid:13)(cid:13)(cid:0)W(cid:48)(cid:48)(cid:48)(φi)φ∆iB1,n
(cid:1)(cid:13)(cid:13)H2 +
(cid:1)(cid:13)(cid:13)H2
+ δ¯λ(cid:13)(cid:13)(cid:0)W(cid:48)(cid:48)(cid:48)(φi)B2,nLiφ∆i
(cid:13)(cid:13)(cid:13)W (k)(un)

+ O(δ2).

and hence the (4.2.20) becomes

(cid:13)(cid:13)(cid:13)(cid:13)−Π0

δI
δu

(Φn(p, ¯λ))

(cid:13)(cid:13)(cid:13)(cid:13)H2

We know that the functions Bj,n ∈ H2 for j = 1, 2 and by the smoothness of W , for k ≤ 4

(cid:13)(cid:13)(cid:13)L∞ ≤ α1. Here the value of α1 depends upon

there exists an α1 > 0 such that
the uniform bound on (cid:107)un(cid:107)L∞. Utilizing all these facts we get upper bound for the H2

norm of -Π0

δI
δu(Φn(p)),

(cid:13)(cid:13)(cid:13)(cid:13)−Π0

δI
δu

(Φn(p, ¯λ))

(cid:13)(cid:13)(cid:13)(cid:13)H2

+(cid:13)(cid:13)W(cid:48)(cid:48)(cid:48)(φi)(cid:13)(cid:13)L∞ (cid:107)φ∆iLiφ∆i(cid:107)H2 +

i=1

i φ∆i

(cid:13)(cid:13)(cid:13)H2

(cid:18)(cid:13)(cid:13)(cid:13)L2
≤ n(cid:88)
+ δ¯λ(cid:13)(cid:13)W(cid:48)(cid:48)(cid:48)(φi)(cid:13)(cid:13)L∞(cid:13)(cid:13)(cid:0)φ∆iB1,n
(cid:1)(cid:13)(cid:13)H2 +
+ δ¯λ(cid:13)(cid:13)W(cid:48)(cid:48)(cid:48)(φi)(cid:13)(cid:13)L∞(cid:13)(cid:13)(cid:0)B2,nLiφ∆i
(cid:1)(cid:13)(cid:13)H2
(cid:18)(cid:13)(cid:13)(cid:13)L2
≤ n(cid:88)

+ α1 (cid:107)φ∆iLiφ∆i(cid:107)H2 +

i φ∆i

(cid:19)

+ O(δ2)

(cid:13)(cid:13)(cid:13)H2
(cid:13)(cid:13)(cid:0)φ∆iB1,n
(cid:1)(cid:13)(cid:13)H2 +
(cid:1)(cid:13)(cid:13)H2
(cid:13)(cid:13)(cid:0)B2,nLiφ∆i

i=1

+ δ¯λα1

+ δ¯λα1

(cid:19)

+ O(δ2).

(4.2.23)

72

Here we use the fact that the product of two H2 functions lies in H2(See Appendix.),

(cid:13)(cid:13)(cid:13)(cid:13)−Π0

δI
δu

(Φn(p, ¯λ))

(cid:13)(cid:13)(cid:13)(cid:13)H2

(cid:18)(cid:13)(cid:13)(cid:13)L2
≤ n(cid:88)

i=1

i φ∆i

(cid:13)(cid:13)(cid:13)H2
(cid:13)(cid:13)B1,n
(cid:13)(cid:13)H2 +
(cid:13)(cid:13)H2 (cid:107)Liφ∆i(cid:107)H2
(cid:13)(cid:13)B2,n

+ δ¯λα1 (cid:107)φ∆i(cid:107)H2
+ δ¯λα1

+ α1 (cid:107)φ∆i(cid:107)H2 (cid:107)Liφ∆i(cid:107)H2 +

(cid:19)

+ O(δ2).

(4.2.24)

H2+2k for some c1 > 0 and obtain

≤ c1 (cid:107)f(cid:107)
(cid:18)
(cid:107)φ∆i(cid:107)H6 + α1 (cid:107)φ∆i(cid:107)H2 (cid:107)Liφ∆i(cid:107)H2 +

(cid:13)(cid:13)H2 +
(cid:13)(cid:13)B1,n
(cid:19)
(cid:13)(cid:13)H2 (cid:107)φ∆i(cid:107)H2

+ δ¯λα1 (cid:107)φ∆i(cid:107)H2
+ δ¯λα1

(cid:13)(cid:13)B2,n

(4.2.25)

+ O(δ2).

We also use the fact that

(cid:13)(cid:13)(cid:13)(cid:13)−Π0

δI
δu

(Φn(p, ¯λ))

nf

(cid:13)(cid:13)(cid:13)Lk
(cid:13)(cid:13)(cid:13)(cid:13)H2

(cid:13)(cid:13)(cid:13)H2
≤ n(cid:88)

i=1

(cid:104) pi−1+pi

2

pi+pi+1

2

,

On the window

(cid:105)

, we write

φ∆i = φmaxe−√
(cid:104) pi−1+pi

pi+pi+1

,

2

2

(cid:105)

for k = 1, 2, . . . on

M > 0 such that

where φmax is the amplitude of the pulse. Using this form of φ∆i, we obtain (cid:107)φ∆i(cid:107)

α−(z−pi−1) + φmaxe−√

α−(pi+1−z),

(4.2.26)

Hk = O(δ)

. Hence, we conclude that there exists a constant

(cid:13)(cid:13)(cid:13)(cid:13)−Π0

δI
δu

(cid:13)(cid:13)(cid:13)(cid:13)H2

(Φn(p, ¯λ))

≤ M δ.

(4.2.27)

(H1) The spectrum of each operator −Π0L2

n(p) consists of a stable part σs and a slow

part σ0. The spectrum of L2

n(p) has been examined in detail in the proof of Lemma (4.1.4).

73

It can be easily seen that σ(−Π0L2

n(p)) = σ(−Π0L2

n(p)Π0) ∪ {0} and further

σ(−Π0L2

n(p)Π0) = σs ∪ σ0,

(4.2.28)

where σs ⊂ {λ|λ ≤ −ks} for some ks > 0 and σ0 ⊂ {λ||λ| ≤ c0e− k0

ε } for some k0, c0 > 0

and σ0 consists of n + 1 eigenvalues.

(H2) Each operator −Π0L2

n(p) generates a C0 semigroup Sp which satisfies

(cid:13)(cid:13)Sp(t)u(cid:13)(cid:13)H2 ≤ M e−kst (cid:107)u(cid:107)H2 ,

(4.2.29)

for all t ≥ 0, u ∈ X⊥

n+1(p). This estimate on the semigroup is a result of Pr¨uss-Gearhart

Theoremwhich states that the boundedness of a C0-semigroup generated by an operator is

from boundedness of the resolvent on the right half plane(See [Gearhart, 1978] and

[Pr¨uss, 1984]). To complete the verification, we need to show the boundedness of the
resolvent of the constrained operator over the space X⊥
constrained to the space X⊥
Pp : H2(R) → Xn+1(p). Introduce the constrained operator Πp(−Π0L2
generating the constrained semigroup onto X⊥
of −Π0L2

n+1(p) because the semigroup is
n+1(p). Let Πp = I − Pp be the orthogonal projection where
n(p)Π0)Πp that is

n(p)Π0, it is easy to see that , σ(Πp(−Π0L2

n+1(p). Since we already obtained the spectrum
n(p)Π0)) \ σ0

n(p)Π0)Πp) = σ((−Π0L2

and hence
σ(Πp(−Π0L2
self-adjoint operator and so, ζ ∈ R(Πp(−Π0L2

n(p)Π0)Πp) ⊂ {λ|λ ≤ −ks}. On the other hand, Πp(−Π0L2

n(p)Π0)Πp) = {z ∈ C|(Πp(−Π0L2

n(p)Π0)Πp is a
n(p)Π0)Πp −

74

ζ) is bijective} and λ ∈ σ(Πp(−Π0L2

n(p)Π0)Πp),

(cid:13)(cid:13)(cid:13)(cid:13)(cid:16)

(cid:17)−1(cid:13)(cid:13)(cid:13)(cid:13) ≤ sup

λ

Πp(−Π0L2

n(p)Π0)Πp − ζ

|λ − ζ|−1,

(4.2.30)

(See [Kato, 1976]).

(H3) As mentioned earlier, here we establish the H2-coercivity of Π0L2
X⊥
n+1(p) which is a result of the H2-coercivity of L2
(4.1.5) that there exists a ˜µ1 > 0, independent of domain size, such that

n(p) over X⊥

n(p) over the space

n (p). Recall from Corollary

(cid:104)L2

n(p)v, v(cid:105) ≥ ˜µ1 (cid:107)v(cid:107)2

H2 ,

for all v ∈ X⊥
Let v ∈ X⊥

n (p).
n (p). Then, w = Π0v ∈ X⊥

n+1(p) ⊂ X⊥

n (p) and

(cid:104)Π0L2

n(p)w, w(cid:105) = (cid:104)Π0L2

n(p)Π0v, Π0v(cid:105)

= (cid:104)L2
= (cid:104)L2
= (cid:104)L2

0v(cid:105)
n(p)Π0v, Π2
n(p)Π0v, Π0v(cid:105)
n(p)w, w(cid:105) ≥ ˜µ1 (cid:107)w(cid:107)2

H2 ,

which proves the H2-coercivity of Π0L2

n(p) over the space X⊥

n+1(p).

(H4) The normalized adjoint eigenvectors {ψ1, ..., ψn+1} of the space Xn+1(p) which are

same with the normalized eigenvectors of the space since L2

n(p) is self-adjoint satisfy

(cid:16)(cid:107)ψi(p)(cid:107)H2 +

max
i=1,...,n
p∈P

(cid:13)(cid:13)(cid:13)∇2

(cid:17) ≤ M,

(cid:13)(cid:13)(cid:13)H2

pψi(p)

(4.2.31)

75

for some M since φ(cid:48)

h(z − pi) ∈ Cn for some n. Analyticity of the eigenvectors of an

unbounded, self-adjoint operator with compact resolvent is proved in details in [Kriegl et al., 2011].

Following the verification of these conditions required in the main theorem in [Promislow, 2002],

we apply the theorem to problem 4.2.1 and obtain the desired result.

76

4.3 Pulse Dynamics

In this section, we study the ODE’s given in (4.2.13) to understand the evolution of an initial
data given in a neighborhood of the manifold Mn.

Reformulating the evolution equations of pulse locations given in [Promislow, 2002] we

obtain n + 1 evolution equations, n equations for the pulse locations and an equation for the

background parameter ¯λ,

i =(cid:10)−Π0

p(cid:48)

δI
δu

(Φn(p, ¯λ)),

h(z − pi)(cid:11) + O(δ2),

φ(cid:48)

1(cid:13)(cid:13)φ(cid:48)
(cid:13)(cid:13)2

h

L2

for i = 1, . . . , n and the evolution of background parameter

¯λ(cid:48) =(cid:10)−Π0

δI
δu

(Φn(p, ¯λ)), 1(cid:11),

(4.3.1)

(4.3.2)

where 1 is the adjoint eigenfunction corresponding to B2,n which is one of the n+1 eigenfunctions
of −Π0

δu(Φn(p, ¯λ)).
δI

Recall that the n-pulse ansatz is given as Φn(p, ¯λ)) = un + δλB2,n with un defined in

(2.2.5). Note that the mass of an n-pulse configuration is

M = nMh +

b−L2

ε

+ δ¯λ

L2
εα2−

,

(4.3.3)

(cid:90) L2

ε

0

(φh − b−)dz is the mass of the homoclinic solution.

where Mh =
Remark 4.3.1. We assume that n is a fixed number, independent of ε, and |pi+1 − pi| ≥ (cid:96)
for all i = 1, . . . , n. These choices provide us δ := e−√

α−(cid:96) (cid:28) εp for all p.

77

Let the initial mass of the polymer added to the solvent be

M0 = nMh +

b−L2

ε

+ ¯M0,

(4.3.4)

where ¯M0 is the excess mass remaining after the formation of n-pulses. Evaluating the inner

product given in (4.3.1) and (4.3.2) we would like to construct explicit ODE’s for the pulse

locations and the background state ¯λ to study their evolutions.

Inserting the expression

obtained in (4.2.17) in the inner product in (4.3.1) we have

(cid:28)

p(cid:48)
i =

− Π0

(cid:18)(cid:16)Li − W(cid:48)(cid:48)(cid:48)(φi)φ∆i

(cid:17)Liφ∆i + δ¯λ

(cid:16)Li − W(cid:48)(cid:48)(cid:48)(φi)φ∆i
(cid:17)LiB2,n+
(cid:19)
(cid:29)
(cid:13)(cid:13)2
1(cid:13)(cid:13)φ(cid:48)
− δ¯λW(cid:48)(cid:48)(cid:48)(φi)B2,nLiφ∆i + O(δ2)

φ(cid:48)

,

.

i

i

L2

(4.3.5)

Remark 4.3.2. Since Π0 is self-adjoint, we project that onto φ(cid:48)
φ(cid:48)
i + O(δ). Hence, the higher order terms in p(cid:48)

i are O(δ2).

i and obtain that Π0φi

(cid:48) =

p(cid:48)
i = −
(cid:28)

= −

(cid:28)(cid:16)Li − W(cid:48)(cid:48)(cid:48)(φi)φ∆i

(cid:17)Liφ∆i + δ¯λ

− δ¯λW(cid:48)(cid:48)(cid:48)(φi)B2,nLiφ∆i,

i φ∆i − W(cid:48)(cid:48)(cid:48)(φi)φ∆iLiφ∆i + δ¯λL2
L2

− δ¯λW(cid:48)(cid:48)(cid:48)(φi)B2,nLiφ∆i,

(cid:16)Li − W(cid:48)(cid:48)(cid:48)(φi)φ∆i
(cid:17)LiB2,n+
(cid:29)
1(cid:13)(cid:13)φ(cid:48)
(cid:13)(cid:13)2
(cid:29)

iB2,n − δ¯λW(cid:48)(cid:48)(cid:48)(φi)φ∆iLiB2,n+

+ O(δ2)

Π0φ(cid:48)

L2

i

i

+ O(δ2).

1(cid:13)(cid:13)φ(cid:48)
(cid:13)(cid:13)2

i

L2

φ(cid:48)

i

(4.3.6)

78

(cid:28)

Here since L2

i is self-adjoint and φ(cid:48)

i ∈ ker(Li), we have (cid:104)L2

i (·), φ(cid:48)

i(cid:105) = 0.

− W(cid:48)(cid:48)(cid:48)(φi)φ∆iLiφ∆i − δ¯λW(cid:48)(cid:48)(cid:48)(φi)φ∆iLiB2,n − δ¯λW(cid:48)(cid:48)(cid:48)(φi)B2,nLiφ∆i,
p(cid:48)
i = −
+ O(δe−√
(cid:28)
W(cid:48)(cid:48)(cid:48)(φi)φ∆iLiφ∆i, φ(cid:48)

(cid:28)
W(cid:48)(cid:48)(cid:48)(φi)(φ∆iLiB2,n + B2,nLiφ∆i),

α−(cid:96))

(cid:29)

+ δ¯λ

=

i

(cid:13)(cid:13)2
1(cid:13)(cid:13)φ(cid:48)

i

L2

(cid:29)

φ(cid:48)

i

L2

1(cid:13)(cid:13)φ(cid:48)
(cid:13)(cid:13)2
(cid:29)

i

φ(cid:48)

i

+ O(δ2).

(4.3.7)

Transposing W(cid:48)(cid:48)(cid:48)(φi) to the right of the inner products and writing W(cid:48)(cid:48)(cid:48)(φi)φ(cid:48)

we obtain

p(cid:48)
i =

(cid:13)(cid:13)2
1(cid:13)(cid:13)φ(cid:48)

i

L2

(cid:28)
φ∆iLiφ∆i,(cid:0)W(cid:48)(cid:48)(φi)(cid:1)
(cid:124)

(cid:123)(cid:122)

z

A

(cid:29)
(cid:125)

+δ¯λ

(cid:13)(cid:13)2
1(cid:13)(cid:13)φ(cid:48)

i

L2

(cid:28)
(cid:124)

+ O(δ2).

φ∆iLiB2,n + B2,nLiφ∆i,(cid:0)W(cid:48)(cid:48)(φi)(cid:1)

i = (W(cid:48)(cid:48)(φi))z,
(cid:29)
(cid:125)

z

(cid:123)(cid:122)

B

For the calculation of A we use the explicit formula of Li,

(cid:68)
(cid:68)

(cid:69)
φ∆iLiφ∆i,(cid:0)W(cid:48)(cid:48)(φi)(cid:1)
(cid:69) −(cid:68)
∆i,(cid:0)W(cid:48)(cid:48)(φi)(cid:1)

φ∆iφ(cid:48)(cid:48)

z

z

A =

=

∆i,(cid:0)W(cid:48)(cid:48)(φi)(cid:1)

z

W(cid:48)(cid:48)(φi)φ2

(4.3.8)

(4.3.9)

(cid:69)

.

(cid:16) 1
Taking W(cid:48)(cid:48)(φi) to the other side in the second inner product and observing that W(cid:48)(cid:48)(φi)(W(cid:48)(cid:48)(φi))z =

(cid:0)W(cid:48)(cid:48)(φi)(cid:1)2(cid:17)

, we obtain

2

z

(cid:68)

∆i,(cid:0)W(cid:48)(cid:48)(φi)(cid:1)

φ∆iφ(cid:48)(cid:48)

z

(cid:69) −(cid:68)

φ2
∆i,

(cid:16)(cid:0)W(cid:48)(cid:48)(φi)(cid:1)2(cid:17)

(cid:69)

.

z

1
2

(4.3.10)

A =

To evaluate these inner products, we write φi = b− + φmaxe−√

α−|z−pi| for all i = 1,··· , n

79

where φmax represents the maximum value of a single pulse in the n-pulse configuration, and
α−(pi+1−z).

we have φ∆i = φmaxe−√

on the window

pi+pi+1

2

(cid:105)

,

2

(cid:104) pi−1+pi
(cid:68)
(φmaxe−√
−(cid:68)
φmaxe−√

A = α−

α−(z−pi−1) + φmaxe−√
α−(z−pi−1) + φmaxe−√

α−(z−pi−1)+φmaxe−√
(cid:69)
α−(pi+1−z))2,(cid:0)W(cid:48)(cid:48)(φi)(cid:1)
(cid:16)(cid:0)W(cid:48)(cid:48)(φi)(cid:1)2(cid:17)

(cid:69)

α−(pi+1−z),

+

z

.

1
2

z

We integrate by parts the terms in the last two lines of (4.3.11) and obtain

√

α−(cid:10)φ2
α−(cid:10)φ2
maxe−2

√
maxe−2
√

A = −2α−
√

+

α−(pi+1−z) − φ2

α−(pi+1−z) − φ2

maxe−2

√
maxe−2
√

α−(z−pi−1), W(cid:48)(cid:48)(φi)(cid:11)+
α−(z−pi−1),(cid:0)W(cid:48)(cid:48)(φi)(cid:1)2(cid:11).

Further, writing the explicit formula for φi and Taylor expanding W(cid:48)(cid:48)(φi) about b− we obtain

(4.3.11)

(4.3.12)

(4.3.13)

(cid:68)

A = − 2α−

√

+

√

= −2α−

α−

(cid:68)

√
α−

+

α−(z−pi−1),

α−(z−pi−1),

α−

√

+

√

√

α−

(cid:68)

α−(pi+1−z) − φ2

√
maxe−2
φ2
W(cid:48)(cid:48)(b− + φmaxe−√
(cid:68)
maxe−2
(cid:16)
φ2
W(cid:48)(cid:48)(b− + φmaxe−√
√
α−(pi+1−z) − φ2
maxe−2
φ2
W(cid:48)(cid:48)(b−) + W(cid:48)(cid:48)(cid:48)(b−)φmaxe−√
√
√
maxe−2
maxe−2
(cid:16)
φ2
W(cid:48)(cid:48)(b−) + W(cid:48)(cid:48)(cid:48)(b−)φmaxe−√

√
maxe−2
(cid:69)
α−|z−pi|)
maxe−2
α−(pi+1−z) − φ2
(cid:17)2(cid:69)
α−|z−pi|)
√
α−(z−pi−1),
maxe−2
α−|z−pi|(cid:69)
α−|z−pi|(cid:17)2(cid:69)

α−(pi+1−z) − φ2

α−(z−pi−1),

+

.

80

Recall that W(cid:48)(cid:48)(b−) = α− and define γ− := W(cid:48)(cid:48)(cid:48)(b−). Rearranging (4.3.13) provides

A = −2α−

√

α−(cid:10)φ2
α−(cid:10)φ2

√

+

√

α−(pi+1−z) − φ2

maxe−2
α− + γ−φmaxe−√

√
α−(pi+1−z) − φ2
maxe−2
α2− + 2α−γ−φmaxe−√

α−(z−pi−1),

√

√

maxe−2
α−|z−pi|(cid:11)+
maxe−2
α−|z−pi| + γ2−φ2
√

α−(z−pi−1),
maxe−2

(4.3.14)

α−|z−pi|(cid:11).

√

√

α− to the right side of

For further simplification we move the constants −2α−

α− and

the inner product.

A =(cid:10)φ2
√
maxe−2
+(cid:10)φ2
maxe−2
√
α−α2− + 2

− 2α2−
√

α−(z−pi−1),

√
α−(pi+1−z) − φ2
maxe−2
√
√
α−γ−φmaxe−√
α− − 2α−
α−(pi+1−z) − φ2
maxe−2
√
α−α−γ−φmaxe−√

α−|z−pi| +

α−(z−pi−1),
√

√

α−|z−pi|(cid:11)+

(4.3.15)

α−|z−pi|(cid:11).

√
maxe−2

α−γ2−φ2

Adding up these two inner products and grouping or canceling common terms we obtain

√
maxe−2
√
− α2−
√

A =(cid:10)φ2
=(cid:10)e−2
(cid:124)
+(cid:10)e−2
(cid:124)

√

α−φ2
√

α− + γ2−

α−(pi+1−z) − e−2

√

I
√
α−(pi+1−z) − e−2

α−(pi+1−z) − φ2

α−(z−pi−1),

α−|z−pi|(cid:11)

√
maxe−2
√
maxe−2
(cid:123)(cid:122)
α−(z−pi−1),−φ2
α−(z−pi−1), γ2−

√

(cid:123)(cid:122)

II

maxα2−

√

α−(cid:11)
(cid:125)
maxe−2

α−φ4

(cid:104) pi−1+pi

2

pi+pi+1

2

,

(cid:105)

(4.3.16)

α−|z−pi|(cid:11)
(cid:125)

.

√

. Let mi =

pi−1+pi

2

and

We evaluate these inner products on the interval

81

mi+1 =

pi+pi+1

2

. We start by calculating the first inner product I,

mi+1(cid:90)
√
(e−2

√

I = −α2−

α−φ2

max

√
α−(pi+1−z) − e−2

α−(z−pi−1))dz

= −α2−φ2
max
2
= −α2−φ2
max
2
+ e−√

mi
√
√
α−(pi+1−z) + e−2
(e−2
(cid:16)
α−(pi+1−pi) − e−√
e−√
α−(pi+1+pi−2pi−1) − e−√

α−(z−pi−1))

α−(2pi+1−pi−pi−1)

α−(pi−pi−1)(cid:17)

.

(cid:12)(cid:12)(cid:12)(cid:12)mi+1

mi

Similarly, we calculate the second inner product and obtain

√

II = γ2−

α−φ4

α−(pi+1−z) − e−2

√

α−(z−pi−1))e−2

√

α−|z−pi|)dz

mi+1(cid:90)
√
(e−2

α−(pi+1−pi−1)+

4

√

max

α−(pi+1−pi)(cid:17)

(cid:16)1
mi
√
(e−2
α−(pi+1−pi) − e−2
√
(pi+1 − pi)e−2
(cid:16)1
α−(pi−pi−1)+
(pi − pi−1)e−2
2
√
√
(e−2
α−(pi−pi−1) − e−2

α−(pi+1−pi−1)(cid:17)

max

√

+

.

max

+

= γ2−φ4
1
2
− γ2−φ4
1
4

+

(4.3.17)

(4.3.18)

Then, adding I and II together and simplifying we obtain

A = −α2−φ2
max
2
+ e−√

(cid:16)
α−(pi+1−pi) − e−√
e−√
α−(pi+1+pi−2pi−1) − e−√
(cid:16)1
α−(pi+1−pi) − 1
4
√
(pi+1 − pi)e−2

√
e−2
α−(pi+1−pi) − 1
2

√
e−2

max

4

+ γ2−φ4
1
2

+

α−(2pi+1−pi−pi−1)

α−(pi−pi−1)(cid:17)

+

α−(pi−pi−1)+
√
(pi − pi−1)e−2

α−(pi−pi−1)(cid:17)

.

(4.3.19)

82

With similar calculations we evaluate the last inner product B. Taking B2,n = α−2− away
from pulses provides φ∆iLiB2,n = −α−2− W(cid:48)(cid:48)(φi)φ∆i and B2,nLiφ∆i = α−2− φ(cid:48)(cid:48)

∆i−α−2− W(cid:48)(cid:48)(φi)φ∆i.

Combining these two in the inner product B making further simplifications we obtain

B =

(cid:69)
(cid:68)
φ∆iLiB2,n + B2,nLiφ∆i,(cid:0)W(cid:48)(cid:48)(φi)(cid:1)
(cid:68)
(cid:68)
(cid:69)
+ α−2−
= −2α−2−
(cid:68)
(cid:68)
∆i,(cid:0)W(cid:48)(cid:48)(φi)(cid:1)
+ α−2−
= −α−2−
φ(cid:48)(cid:48)

W(cid:48)(cid:48)(φi)φ∆i,(cid:0)W(cid:48)(cid:48)(φi)(cid:1)
(W(cid:48)(cid:48)(φi))2(cid:17)
(cid:16)
(cid:69)

∆i,(cid:0)W(cid:48)(cid:48)(φi)(cid:1)
(cid:69)

φ∆i,

φ(cid:48)(cid:48)

z

z

.

z

z

(cid:69)

z

(4.3.20)

Here recall that φ(cid:48)(cid:48)(cid:48)

∆i = α−φ(cid:48)

∆i. Expanding W(cid:48)(cid:48)(φi) about b− and grouping common terms

in these two inner products give

B = α−2−
= α−2−

(cid:68)
(cid:68)
∆i, (W(cid:48)(cid:48)(φi))2(cid:69) − α−1−
φ(cid:48)
φ(cid:48)
(cid:68)
∆i, α2− + 2γ−α−φmaxe−√
φ(cid:48)
(cid:68)
−√
α−|z−pi|
− α−1−
φ(cid:48)
α−|z−pi|(cid:69)
∆i, α− + γ
e
∆i, e−√
= α−1− γ−φmax
φ(cid:48)

∆i,(cid:0)W(cid:48)(cid:48)(φi)(cid:1)(cid:69)
α−|z−pi| + γ2−φ2
maxe−2
(cid:69)
(cid:68)
+ α−2− γ2−φ2

(cid:68)

max

√
∆i, e−2
φ(cid:48)

√

α−|z−pi|(cid:69)
α−|z−pi|(cid:69)

(4.3.21)

.

Then, we calculate the inner products in B.

mi+1(cid:90)

B =

α−1− γ−φmax

√

α−(e−√

α−(pi+1−z) − e−√

α−(z−pi−1))e−√

α−|z−pi|+

mi
+ α−2− γ2−φ2

max

√
α−(e−√

α−(pi+1−z) − e−√

√
α−(z−pi−1))e−2

α−|z−pi|dz

(4.3.22)

83

=

1
2

+

3

√

max

max

α−(pi+1−pi−1))
α−(pi−pi−1))+
− 1
2

max(e−√
α−(pi+1−pi) − e−√
√
α−(pi − pi−1)e−√
(cid:16)1
(e−√
α−(pi+1−pi) − e
(cid:17)
α−(pi−pi−1))
− 3
α−(pi−pi−1) − e
+
2
max(e−√
α−(pi+1−pi−1) − e−√
α−(pi−pi−1))
√
α−(pi+1 − pi)e−√
(cid:16)1
− 1
2

α−(pi+1−pi))+
α−(3pi+1−pi−2pi−1) − e−√

max

√

√

(e

α−1− γ−φ2
α−1− γ−φ2
− 1
2
+ α−2− γ2−φ3
− (e−√
α−1− γ−φ2
α−1− γ−φ2
+ α−2− γ2−φ3
− (e

− 3
2

1
2

1
2

√

+

(cid:17)
α−(pi+1−pi))

.

max

3

α−(pi+1−pi) − e−√

α−(2pi+1+pi−3pi−1))+

(4.3.23)

α−(pi−pi−1))+

Inserting (4.3.19), and (4.3.23) in (4.3.8) we obtain the evolution equations of pulse locations,

(cid:16)

p(cid:48)
i =

(cid:18)

1(cid:13)(cid:13)φ(cid:48)
(cid:13)(cid:13)2
+ e−√

L2

i

+ γ2−φ4
1
2

+

+

α−(2pi+1−pi−pi−1)

4

+

max

e−√

√
e−2

α−(pi+1−pi) − e−√
α−(pi−pi−1)(cid:17)

α−(pi−pi−1)+
√
(pi − pi−1)e−2
α−(pi+1−pi) − e−√

−α2−φ2
max
2
α−(pi+1+pi−2pi−1) − e−√
(cid:16)1
√
α−(pi+1−pi) − 1
e−2
4
√
(pi+1 − pi)e−2
α−(pi+1−pi) − 1
(cid:16)
(cid:16)1
δ¯λ(cid:13)(cid:13)φ(cid:48)
(cid:13)(cid:13)2
2
(e−√
α−1− γ−φ2
α−((pi+1 − pi)e−√
α−(pi+1−pi) − (pi − pi−1)e−√
(cid:16)1
α−(pi+1−pi) − e−√
(e−√
α−(pi−pi−1)) − (e
1
3

+ α−2− γ2−φ3
− e−√

α−(3pi+1−pi−2pi−1) − e

α−(pi+1−pi) − e

− 1
2

− 3
2

− 1
2

√

max

max

L2

√

√

√

(e

+

+

2

3

i

+

α−(pi−pi−1)(cid:17)(cid:19)
α−(pi−pi−1)(cid:17)
α−(pi−pi−1)) + (e−√

α−(pi−pi−1))+

+

√

− 3
2

α−(pi−pi−1))+
(cid:17)
α−(2pi+1+pi−3pi−1))

α−(pi+1−pi)+

.

(4.3.24)

Observing that(cid:13)(cid:13)φ(cid:48)
the big terms e−√

(cid:13)(cid:13)2

√
φ2
max
2

α−

i

L2 =

and neglecting exponentially small terms comparing to
α−(cid:96)i where (cid:96)i represents the distance between the centers of two adjacent

84

pulses we conclude that

p(cid:48)
i = −α

3/2−

(cid:16)

e−√

α−(pi+1−pi) − e−√

α−(pi−pi−1)(cid:17)

+ O(δ2),

(4.3.25)

assuming p0 = −p1 and pn+1 = 2

ε − pn.
L2

After we derived the ODE’s for pulse locations we also construct the ODE for the

background parameter ¯λ evaluating the inner product in (4.3.2).

¯λ(cid:48) =(cid:10)−Π0

δI
δu

(Φn(p, ¯λ)), 1(cid:11).

(4.3.26)

Since Π0 is self-adjoint and Ker(Π0) = span{1}, we move Π0 to the right side of the inner

product and obtain

¯λ(cid:48) = 0.

(4.3.27)

From this we conclude that there are no background dynamics. Hence, the ODE system,

consisting of n + 1 evolution equations for n pulse positions and ¯λ, is

¯λ(cid:48) = 0,
p(cid:48)
i = −α

3/2−

(cid:16)

e−√

α−(pi+1−pi) − e−√

α−(pi−pi−1)(cid:17)

+ O(δ2).

(4.3.28)

It is easy to see that if the pulses are equally separated, pi+1 − pi = pi − pi−1 = (cid:96), for (cid:96) > 0

big enough for all i = 1, . . . , n then, these pulses are stationary, namely,

p(cid:48)
i = 0,

∀i = 1,··· , n.

(4.3.29)

Further, we would like to study the stability of these stationary n-pulse configurations

85

examining the Jacobian matrix of this ODE system.

2

0
γ − γ

2

γ − γ
− γ
0 − γ
...

2

0
...

0

. . .

0

. . .

0

2

0
γ − γ
2
. . .
. . .

. . .

. . .

0
. . .
0
. . . − γ

2

0
...

γ

. . .
0 − γ

2



(4.3.30)



J =

0

0

(cid:18)

where γ := −α2−e−√

α−(cid:96). The matrix J has n eigenvalues

λk = −α2−

1 + cos

e−√

α−(cid:96) < 0,

(4.3.31)

(cid:18) k

(cid:19)(cid:19)

n + 1

for all k = 1, . . . , n. From this, we conclude that the stationary solutions are spectrally

stable.

Moreover, assuming that the initial data is given for two-pulse configuration U0 = φ1 +φ2

and setting (cid:96)i := pi − pi−1 we obtain the evolution equations for pulse locations as

(cid:16)
e−√
e−√
α−(cid:96)3 − e−√

α−(cid:96)2 − e−√
(cid:17)
α−(cid:96)2

max

(cid:16)

p(cid:48)
1 = −α
p(cid:48)
2 = −α

3/2− φ2

3/2−

(cid:17)

α−(cid:96)1

+ O(δ2),

+ O(δ2).

(4.3.32)

If the distance between φ1 and φ2 is smaller than their distance to adjacent pulses, (cid:96)3 >
(cid:96)1 > (cid:96)2 or (cid:96)1 > (cid:96)3 > (cid:96)2, then p(cid:48)

2 > 0. From this analysis, we conclude that if

1 < 0 and p(cid:48)

two adjacent pulses are closer to each other than other neighbor pulses, then they repel each

other. If two pulses more far apart comparing to other neighbor pulses than they attract

86

each other.

4.4 Conclusion

We have shown that the pulses are attracted into and remain within an O(δ2) window of the

equally spaced (periodic) distribution. Moreover, the full solution remains within an O(δ)

neigborhood in H2 of the periodic n-pulse. By compactness, a subsequence of times tn with
tn → ∞ as n → ∞ exists such that u(tn) converges to u∗ in the ball of radius δ of the
n-periodic solution. As this is a gradient flow, u∗ must be an equilibrium and then we must
have u(t) → u∗ for the whole sequence as traversing the distance between u∗ and a distinct

equilibrium value infinitely many times would expend infinite energy. In particular, the flow

converges to an equilibrium which is exponentially close to the periodic n-pulse.

87

APPENDIX

88

Appendix

A bound for H 2 norm of the product of two H 2 functions

Lemma A.0.1. Let f, g ∈ H2([R]). Then, there exists a constant C > 0 such that

Proof.

(cid:107)f g(cid:107)H2 =

=

=

(1 + k2)| ˆf (k − ˜k)||ˆg(˜k)|d˜k

(cid:107)f g(cid:107)H2 ≤ C (cid:107)f(cid:107)H2 (cid:107)g(cid:107)H2 .
(cid:90)
(1 + k2)2|(cid:99)f g(k)|2dk,
(cid:90)
(cid:18)(cid:90)
(cid:90)

(1 + k2)2

(cid:18)(cid:90)

R

R

| ˆf (k − ˜k)ˆg(˜k)|d˜k

R

(A.0.1)

(A.0.2)

dk,

dk.

(cid:19)2
(cid:19)2

R

R

89

Writing k = k − ˜k + ˜k and then by the inequality (a + b)2 ≤ 2a2 + 2b2, we have

(cid:107)f g(cid:107)H2 ≤

(1 + (k − ˜k)2) ˆf (k − ˜k)||ˆg(˜k)|d˜k + (1 + ˜k2)| ˆf (k − ˜k)||ˆg(˜k)|d˜k

(cid:17)| ˆf (k − ˜k)||ˆg(˜k)|d˜k

(cid:19)2

dk,

(cid:90)

R

R

R

22(1 + (k − ˜k)2) + 22(1 + ˜k2)

(cid:18)(cid:90)
(cid:16)
(cid:90)
(cid:90)
(cid:16)
(cid:90)
(cid:16)|(1 + k2) ˆf (k)| ∗ |ˆg(k))| + |(1 + k2)ˆg| ∗ | ˆf|(cid:17)2
(cid:90)
(cid:16)|(1 + k2) ˆf (k)| ∗ |ˆg(k))|(cid:17)2
(cid:19)
(cid:18)(cid:13)(cid:13)(cid:13)|(1 + k2) ˆf| ∗ |ˆg)
(cid:13)(cid:13)(cid:13)|(1 + k2)ˆg| ∗ | ˆf )|(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)2

dk + 26

(cid:90)

+

R

R

R

R

L2

.

L2

= 24

= 24

≤ 26

= 26

dk,

(cid:16)|(1 + k2)ˆg| ∗ | ˆf )|(cid:17)2

dk,

(cid:17)2

dk,

(A.0.3)
By the Young’s inequality for convolutions that is (cid:107)f ∗ g(cid:107)r ≤ (cid:107)f(cid:107)p (cid:107)g(cid:107)q for f ∈ Lp and

g ∈ Lq with 1

p + 1

q = 1

r + 1,

(cid:107)f g(cid:107)H2 ≤ 26
≤ 26

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13)(1 + k2)ˆg
(cid:19)
(cid:13)(cid:13)(cid:13) ˆf
(cid:13)(cid:13)(cid:13)2

,

L1

L2

(cid:19)

(cid:13)(cid:13)(cid:13)2
(cid:13)(cid:13)(cid:13) ˆf

L1

,

(A.0.4)

(cid:107)ˆg(cid:107)2

L1 +

L2

L1 + (cid:107)g(cid:107)2

H2

(cid:18)(cid:13)(cid:13)(cid:13)(1 + k2) ˆf
(cid:13)(cid:13)(cid:13)2
(cid:18)
(cid:107)f(cid:107)H2 (cid:107)ˆg(cid:107)2
H2 (cid:107)g(cid:107)2

≤ C (cid:107)f(cid:107)2

H2 ,
since (cid:107)g(cid:107)L1 ≤ c(cid:107)g(cid:107)H2 for a constant c > 0.

Second variation of I

The second variation of I with respect to L2 inner product is calculated taking derivative of

i(τ ) twice,

90

i(cid:48)(cid:48)(τ ) =

=

=2

dz,

ε

ε

ε

0

0

d
dτ

d2
dτ 2

(cid:16)
(cid:16)

z (Φn + τ v) − W(cid:48) (Φn + τ v)
∂2

(cid:17)2
z (Φn + τ v) − W(cid:48)(Φn + τ v)
∂2
(cid:17)(cid:16)

(cid:90) L2
(cid:90) L2
(cid:90) L2
(cid:17)
(cid:17)(cid:16)
(cid:16)
z v − W(cid:48)(cid:48) (Φn + τ v) v
z v − W(cid:48)(cid:48) (Φn + τ v) v
(cid:16)
(cid:17)(cid:16)−W(cid:48)(cid:48)(cid:48) (Φn + τ v) v2(cid:17)
∂2
∂2
z (Φn + τ v) − W(cid:48) (Φn + τ v)
(cid:90) L2
(cid:17)(cid:17)2
(cid:16)(cid:16)
∂2
z v − W(cid:48)(cid:48) (Φn + τ v) v
(cid:17)(cid:16)−W(cid:48)(cid:48)(cid:48) (Φn + τ v) v2(cid:17)
(cid:16)
∂2
z (Φn + τ v) − W(cid:48) (Φn + τ v)
∂2

0

0

+

+

ε

=2

dz,

dz.

z v − W(cid:48)(cid:48) (Φn + τ v) v
∂2

(cid:17)

dz,

(A.0.5)

Evaluating this derivative at τ = 0,

(cid:90) L2
(cid:90) L2

0

ε

ε

0

(cid:16)
(cid:16)

z v − W(cid:48)(cid:48) (Φn) v
∂2
(cid:17)2
z − W(cid:48)(cid:48) (Φn)
∂2

(cid:17)2

+

v2 +

i(cid:48)(cid:48)(0) = 2

= 2

(cid:16)
(cid:16)

(cid:17)(cid:16)−W(cid:48)(cid:48)(cid:48) (Φn) v2(cid:17)
(cid:17)(cid:16)−W(cid:48)(cid:48)(cid:48) (Φn) v2(cid:17)

z Φn − W(cid:48) (Φn)
∂2

z Φn − W(cid:48) (Φn)
∂2

dz,

dz.

(A.0.6)

When we insert the expansion for Φn in (A.0.6) and expand it about φn, we have

91

(cid:90) L2
(cid:90) L2

0

ε

ε

0

i(cid:48)(cid:48)(0) =2

=2

dz,

v2+

v2 +

(cid:17)2
z − W(cid:48)(cid:48) (Φn)
∂2

(cid:16)
(cid:17)(cid:16)−W(cid:48)(cid:48)(cid:48) (Φn) v2(cid:17)
(cid:16)
z Φn − W(cid:48) (Φn)
∂2
(cid:1)(cid:17)2
(cid:16)
z − W(cid:48)(cid:48)(cid:0)φn + ελB2,n
(cid:1)(cid:17)(cid:0)−W(cid:48)(cid:48)(cid:48)(cid:0)φn + ελB2,n
(cid:16)
(cid:1) − W(cid:48)(cid:0)φn + ελB2,n
(cid:0)φn + ελB2,n
(cid:1)(cid:17)2
(cid:16)
z −(cid:0)W(cid:48)(cid:48) (φn) + ελW(cid:48)(cid:48)(cid:48) (φn)B2,n
(cid:1)(cid:17)(cid:16)
zB2,n −(cid:0)W(cid:48) (φn) + ελW(cid:48)(cid:48) (φn)B2,n
W(cid:48)(cid:48)(cid:48) (φn) + ελW (4) (φn)B2,n
(cid:18)(cid:16)
(cid:17)(cid:0)W(cid:48)(cid:48)(cid:48) (φn)(cid:1) + O(ε)

(cid:1)(cid:1) v2dz,

(cid:17)2 −(cid:16)

z φn − W(cid:48) (φn)
∂2

z − W(cid:48)(cid:48) (φn)
∂2

v2−

v2dz.

(cid:19)

∂2

∂2
z

∂2

=2

+

ε

(cid:90) L2
(cid:16)
(cid:90) L2

ε

=2

0

0
∂2
z φn + ελ∂2

Since φn solves ∂2

z u − W(cid:48) (u) = 0, at the leading order we obtain

(cid:17)2

z − W(cid:48)(cid:48) (φn)
∂2

v2dz,

(cid:16)

(cid:90) L2
(cid:90) L2

0

ε

ε

0

i(cid:48)(cid:48)(0) = 2

= 2

(Lnv)2 dz,

= 2(cid:104)Ln

2v, v(cid:105).

Thus, the second variation of I at the leading order is

δ2I
δU 2 (Φn) := L2

n.

(cid:17)

v2dz,

(A.0.7)

(A.0.8)

(A.0.9)

H 2 norm of the projection Π0

Lemma A.0.2. Let Π0 be the orthogonal projection of H2 onto U ⊂ H2 given in (4.2.2).
Then, for any u ∈ H2

92

Proof. For any u ∈ H2,

(cid:107)Π0u(cid:107)H2 = (cid:107)u(cid:107)H2 .

(cid:107)Π0u(cid:107)2

H2 = (cid:104)Π0u, Π0u(cid:105)H2

0u, u(cid:105)H2

= (cid:104)Π2
≤ (cid:107)Π0u(cid:107)H2 (cid:107)u(cid:107)H2 ,

(A.0.10)

(A.0.11)

and therefore, (cid:107)Π0u(cid:107)H2 ≤ (cid:107)u(cid:107)H2. On the other hand, since Π0 is a projection we can write
u = Π0u + u(cid:48) where u(cid:48) ∈ U⊥ and obtain

H2 +(cid:13)(cid:13)u(cid:48)(cid:13)(cid:13)2

H2 ,

(cid:107)u(cid:107)2

H2 = (cid:104)Π0u + u(cid:48), Π0u + u(cid:48)(cid:105) = (cid:107)Π0u(cid:107)2

(A.0.12)

which proves that (cid:107)Π0u(cid:107)H2 ≥ (cid:107)u(cid:107)H2.

93

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