NONLINEAR ADIABATIC STABILITY FOR A GENERALIZED REACTION-DIFFUSION SYSTEM By Thomas James Bellsky A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Mathematics 2011 ABSTRACT NONLINEAR ADIABATIC STABILITY FOR A GENERALIZED REACTION-DIFFUSION SYSTEM By Thomas James Bellsky We examine a singularly perturbed, coupled, weakly damped, reaction-diffusion system in one space dimension. This system is examined in the semi-strong pulse interaction regime. We rigorously construct a slow manifold of N -pulse solutions. We identify neutral modes and uncouple them. We solve this reduced nonlinear N -dimensional system with a fixed point method, which generates an equilibrium solution for the reduced system. We turn the coupling back on and continue the slow manifold back to the original system. After analyzing the eigenvalue problem and using renormalization group methods, we show the approximate invariant manifold for the full system is adiabatically stable. We also derive an explicit formula for the pulse dynamics. This work is the first rigorous analysis of the weakly damped regime, in which the essential spectrum approaches the origin. To my family. You are the best—. iii ACKNOWLEDGMENT No road is long with good company. - old proverb First and foremost I want to thank my advisor, Dr. Keith Promislow. Keith has been always candid and enthusiastic in his advice and teaching. I am grateful for the abundance of time he committed to nurturing me into an independent researcher. I have never met a more assiduous mathematician, which by example, has led me to be more so. His cheerful nature has made it an absolute pleasure to work together these past three years. I am thankful to my committee members, whose time and advice has been valuable to my research, and in general, to the professors of the mathematics department at Michigan State University who have provided me with good instruction. I would also would like to acknowledge Dr. Arjen Doelman of Leiden University and Dr. Tasso Kaper of Boston University, who helped formulate the topic of this dissertation and offered guidance. The endeavor of graduate school would never have been possible without the help of my fellow graduate students. The beginning would have been unbearable without study groups with people such as Jim Freitag, Clark Musselman, Joe Timmer, and Jacqueline Dresch. The guidance of older graduate students including Mike Dabkowski and Matt O’Toole was also very helpful. I am very thankful to my brothiv ers in arms Greg Hayrapetyan and Yang Li, as we have grown together under the tutelage of our advisor. Most importantly, I would like to thank my friends and family for their support throughout this process. Their love, support, encouragement, and company has been invaluable to me. I am truly fortunate to have parents that have always advocated the pursuit of knowledge. Finally, I would like to offer gratitude to the family of Bob Hoyer, my sixth grade teacher. I vividly remember the day he introduced me to the notion of doing mathematics for a living, then a novel thought to a wide-eyed eleven year old. v TABLE OF CONTENTS List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . viii 1 Introduction 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 24 2 Construction of an N -pulse Invariant Manifold. 2.1 Explicit form for Φ2 . . . . . . . . . . . . . . . . 2.2 Explicit form for Φ1 . . . . . . . . . . . . . . . . 2.3 Construction of the pulse amplitudes q(p) . . . . 2.4 Existence of an invariant manifold . . . . . . . . . 2.5 Numerical Results . . . . . . . . . . . . . . . . . . 26 32 38 39 50 52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Original System: Ansatz and Residual Estimates 56 3.1 Linear estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Residual estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4 Linearized Equation and Spectrum 4.1 The reduced linearization . . . . . . 4.2 The point spectrum . . . . . . . . . 4.3 Finite rank spectrum . . . . . . . . 4.4 Adjoint eigenfunction estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 78 80 90 91 5 Resolvent and Semigroup Estimates 98 5.1 Spectral projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2 Resolvent estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.3 Semigroup estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6 Nonlinear Adiabatic Stability by Renormalization 6.1 Projected equations . . . . . . . . . . . . . . . . . . 6.2 Decay of the remainder . . . . . . . . . . . . . . . . 6.3 The renormalization group iteration . . . . . . . . . vi Group 118 . . . . . . . . . . 123 . . . . . . . . . . 126 . . . . . . . . . . 155 6.4 Long-time asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7 Future Work 166 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 vii LIST OF FIGURES 1.1 1.2 1.3 1.4 1.5 A typical quasi-steady solution structure for the coupled system (1.1) and (1.2). The V component is localized at the pulse positions pj . The U component has an approximately constant value, qj , on the narrow pulse intervals, and is slowly varying in between the pulses. For interpretation of the references to color in this and all other figures, the reader is referred to the electronic version of this dissertation. . . 2 This figure illustrates the spectral decomposition of the reduced linearization for N = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 This is an illustration of the evolution of the finite rank spectrum for the 2-pulse regularized Gierer-Meinhardt system, for a variety of pulse positions. There are four finite rank eigenvalues. As the pulse separation approaches +∞, the finite rank eigenvalues reside at the left-most point of the loops, corresponding to the weak regime. As the pulse separation decreases, the finite rank eigenvalues separate, one traversing the loop, and the other approaching the real axis, colliding with its complex conjugate, and splitting into a real pair with one approaching the origin and the other retreating towards the essential spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 This figure illustrates, for N = 2 and p1 < p2 , the minimum pulse separation l0 | ln | and also the pulse classes Ktight , Kd , and Kweak . 15 This illustrates the second component pulse paired with the partition of unity χj at three pulse locations. . . . . . . . . . . . . . . . . . . . 20 viii 1.6 This figure generically illustrates the quadratic inequality (1.34) for α < 2/3, so that r1 > 0 and the remainder can be appropriately bounded. Our analysis reduces the size of the remainder to a quadratic inequality, so either the remainder starts smaller than r1 and stays small or it begins larger the r2 . The middle interval (r1 , r2 ) is forbidden. 22 1.7 This figure illustrates the renormalization group technique. The initial condition is decomposed as U0 = Φp0 + W0 (0), the linearization and associated spectral projections are frozen for a time interval sufficient to give decay, but not so long that the secular growth swamps the error. At the end of each renormalization interval, U (τ1 ) = Φ (p(τ1 ))+W (τ1 ) is reprojected into U (τ1 ) = Φ (p1 ) + W1 , where W1 ∈ Xp . The pro1 cess is iterated and the transient associated to the initial perturbation decays to the level of the accuracy of the approximate adiabatic solution. 23 2.1 Notice the function K(V / 2 ) exponentially decays for V >> 2 /2. . 31 2.2 An illustration of the homoclinic orbit of φj . . . . . . . . . . . . . . . 35 2.3 This is a cartoon of the pulse shape of Φ2 at the pulse position pj . Notice how it is localized. . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 This is an illustration of S in two space dimensions. . . . . . . . . . . 47 2.5 This figure is a graph of the solution to (2.104) without the K term in blue and the full solution to (2.104) in red. The solution to the full equation decays faster. . . . . . . . . . . . . . . . . . . . . . . . . . . 53 The fact that the solution to the full equation decays faster is more evident in this semi-log plot in the vertical coordinate. Again, the solution to (2.104) absent the K term is in blue and the solution to the full equation (2.104) is in red. . . . . . . . . . . . . . . . . . . . . 53 2.6 ix 2.7 Each color is one component of the pulse amplitude, where pulse separation is varied which results in a change in amplitude. The horizontal axis is a log-scale of the separation between pulses, continued from a well-separated regime to a semi-strong regime. The values given are |p2 − p1 |. The vertical axis is pulse amplitude. . . . . . . . . . . . . . 54 These two graphs illustrate the three pulses and their amplitudes for |p2 − p1 | = 33 on the left and |p2 − p1 | = 1.6 on the right, corresponding to the previous figure. . . . . . . . . . . . . . . . . . . . . . . . . 55 ˜ This illustrates the point spectrum of L22 that is either positive or near zero. There are N eigenvalues within O( r ) of both λ1 = 0 and λ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.1 This is an illustration of our contour C. . . . . . . . . . . . . . . . . . 99 6.1 This illustrates the quadratic function g(r), where either the remainder starts smaller than r1 and stays small or it begins larger the r2 . The middle interval (r1 , r2 ) is forbidden. . . . . . . . . . . . . . . . . 153 2.8 4.1 x Chapter 1 Introduction The study of self-organizing pattern formation was first considered by Turing [31]. Systems of recent interest are the Gierer-Meinhardt model [18] and the Gray-Scott model [22]. These reaction-diffusion systems consist of activator components which drive pattern formation and inhibitor components which curtail the reaction. They model a variety of chemical reactions, including morphogenesis. We study two classes of systems, the first encompassing a class of singularly perturbed reaction-diffusion equations Ut = −2 U − α µU + −β U α11 V α12 , xx Vt = Vxx − V + U α21 V α22 , 1 (1.1a) (1.1b) with α11 , α21 ≥ 0 and µ > 0, α ≥ 0, β ≥ 0, α12 ≥ 2 and α22 ≥ 2. We study this system for 0 < 1, which introduces the novelty of an asymptotically small linear damping for the first component U . Figure 1.1: A typical quasi-steady solution structure for the coupled system (1.1) and (1.2). The V component is localized at the pulse positions pj . The U component has an approximately constant value, qj , on the narrow pulse intervals, and is slowly varying in between the pulses. For interpretation of the references to color in this and all other figures, the reader is referred to the electronic version of this dissertation. The second class of systems we consider is referred to as an activator-inhibitor system, where the activator component, V , drives the reaction while the inhibitor component, U , curtails production of V . Many references in the literature, including [18], have modeled these systems with an equation of the form (1.1), with α21 < 0. However, the singularity in the nonlinear term at U = 0 suggests unlimited production of the activator when the inhibitor is absent, which has no correspondence 2 to chemical reality. To be consistent with the chemical literature, see for example Chapter 26 of [1], we truncate the singularity and rewrite (1.1) as Ut = −2 U − α µU + −β U α11 V α12 , xx Vt = Vxx − V + κ(U )α21 V α22 , (1.2a) (1.2b) where the function κ is defined as κ(s) =    s s > 2δ    , (1.3)    δ 0 2/3 would require an extension of the approximate invariant manifold to include parts of the essential spectrum, which would be manifested as self-like structures (small in L∞ , large in L1 ) in the localized component V . However, we do not conduct analysis of the α > 2/3 case in 7 this thesis. This work contains the first rigorous construction of an N -pulse adiabatic manifold in the semi-strong regime. We identify the neutral modes of the system (1.4) and identify sufficient conditions under which they may be uncoupled at leading order to obtain a reduced formulation of the system. More specifically, we introduce the pulse positions p(t) ∈ Kl ⊂ RN and the pulse amplitudes q(t) ∈ RN , which evolve 0 with respect to time. The set Kl is defined as 0 Kl = p |pi − pj | > l0 | ln |, l0 > 0, ∀i = j, 1 ≤ i, j ≤ N , 0 (1.9) with l0 sufficiently large so that the localized pulse overlap is O( r ) for r ≥ 2. The N -pulse adiabatic solution is a function of the pulse positions p ∈ RN , where the j-th localized pulse in the V component is centered at pj for j = 1, · · · , N . The pulse positions are well-ordered, so that pi < pj for i < j. The amplitude of the delocalized component U at each pulse position pj is denoted by qj , so U (pj ) = qj for j = 1, · · · , N (see Figure 1.1). For fixed pulse positions p, we rigorously determine a self-consistent mean field for these N -pulse amplitudes q = q(p) by a fixed point method. Using the implicit function theorem, we are able to generate the amplitudes q as local smooth functions of p and > 0. For each p ∈ Kl there exists at least one branch q(p), but often 0 more then one. Also, there exists both a uniform lower and upper bound for at 8 least one branch q(p). For any branch with this uniform bound above and below, we can ignore the δ term in (1.3) (thus contained in the system (1.4)) so long as δ is chosen smaller then the uniform lower bound. We also formulate a non-bifurcation condition, that when met, will guarantee the local persistence of a particular branch q(p). Associated to each branch q(p), we construct an adiabatic N -pulse solution to (1.4) of the form    Φ1 (x; p(t), q(t))  Φp(t) (x) =  , Φ2 (x; p(t), q(t)) (1.10) where Φ1 corresponds to the U component and Φ2 corresponds to the V component (see Section 2.1 and Section 2.2 for the explicit construction of Φp(t) ). Under appropriate restrictions which we detail below, the N -pulse adiabatic solution Φp serves as an adiabatic manifold with boundary for the system, generating a slow flow. Linearizing the full system (1.4) about the adiabatic N -pulse solution generates the linearized operator Lp(t) = L (p(t)) . The heart of the technical elements of this thesis is a detailed analysis of the linearized operator and the associated semigroup. A ˜ ˜ key step is the identification of a reduced linearization Lp(t) = L (p(t)) . In particular, ˜ we show that there exists a ν > 0 independent of such that σ L ∩ {Re(λ) > −ν} can be decomposed into three parts. The first part is the essential spectrum, independent of pulse position p. The essential spectrum consists of the set (−ν, − α µ], which lies within the left-half complex plane. Recall, the asymptotically small damp9 ing term results in an essential spectrum which is asymptotically close to the origin. The second part of the spectrum consists of N -point spectra, σ0 , which corresponds to the translational modes of the localized pulses and whose eigenmodes lie in the tangent plane of the manifold at leading order. By an appropriate restriction on l0 , which controls the localized pulse separation, we can restrict these eigenvalues to reside within O( r ) of the origin, for any r ≥ 2 that we desire. Figure 1.2: This figure illustrates the spectral decomposition of the reduced linearization for N = 2. We show that the remainder of the spectrum, which we call the finite rank spectrum, σf r , can be characterized as solutions of N algebraic equations which we associate to the finite rank potentials in the reduced linearization. The finite rank spectrum evolves at leading order as the pulse positions evolve, and a key issue of this work is reducing the location of the spectral set σf r to an explicit set of algebraic equations. The following figure illustrates the evolution of the spectral set σf r for a 2-pulse solution to the regularized Gierer-Meinhardt system in [12] over a wide range 10    p1  of values of the localized two-pulse positions p =  . p2 Figure 1.3: This is an illustration of the evolution of the finite rank spectrum for the 2-pulse regularized Gierer-Meinhardt system, for a variety of pulse positions. There are four finite rank eigenvalues. As the pulse separation approaches +∞, the finite rank eigenvalues reside at the left-most point of the loops, corresponding to the weak regime. As the pulse separation decreases, the finite rank eigenvalues separate, one traversing the loop, and the other approaching the real axis, colliding with its complex conjugate, and splitting into a real pair with one approaching the origin and the other retreating towards the essential spectrum. A novel feature of our analysis is the identification of a bifurcation parameter θ ≡ α11 − α12 α21 /(α22 − 1), 11 (1.11) which balances the exponents of the nonlinear terms. We demonstrate the existence of the N -pulse manifold for θ = 1, and numerically we observe a unique nontrivial N -pulse solution for θ < 1, while for θ > 1, we typically find multiple (up to 2N ) N -pulse solutions. Enumerating all of the branches, for each branch q i = q i (p) of the N -pulse solution we define the adiabatic manifold of N -pulse solutions as Mi = Φ ·; p, q i (p) , p ∈ K . (1.12) where the set of admissible pulse positions K ⊂ RN is defined as i K = Ki ≡ Kl ∩ Kν , 0 (1.13) i for Kl previously defined and Kν defined as 0 i ˜ Kν = p max Re σf r Lp < −ν . (1.14) i The set Kν imposes an explicit stability condition, which localizes the finite rank spectrum in the left-half complex plane. The stability condition not only rules out potential Hopf type bifurcation, in which N -pulses become unstable to oscillatory modes, but also saddle-node type bifurcation, in which a single N -pulse separates into two (or more) distinct N -pulse solutions. The stability condition is defined in 12 terms of eigenvalues of an explicit N × N matrix, in general the exact nature of i i the set Kν depends sensitively upon the specific system studied. When p ∈ Kν the finite rank spectrum will never approach the origin, so there can be no splitting of an amplitude solution q i , so as a consequence, the non-bifurcation condition previously mentioned is enforced. In order to discuss the possible pulse configurations contained in the set K, we also introduce the set Kweak = p ∈ Rn −1−α/2 pi+1 − pi . (1.15) For our system (1.4), the set Kweak corresponds to the weak interaction regime, described earlier. There exists a unique non-degenerate N -pulse solution in the weak regime which consists of N well-separated copies of the 1-pulse. Since both localized and delocalized components are well-separated, the point spectrum in the weak regime consists of N exponentially close copies of the point spectrum of the 1˜ pulse. If there exists ν > 0 such that the reduced linearization L1 about this 1-pulse satisfies ˜ σ L1 \ {0} ⊂ λ Re(λ) < −ν , (1.16) then Kweak ⊂ K. A second regime is the tight regime in which all the localized pulses, while still well-separated, are crowded into a region over which the delocalized 13 component is asymptotically constant. The set Ktight is defined as Ktight = p ∈ Rn l0 | ln | < pN − p1 −1−α/2 . (1.17) There can be multiple N -pulse branches in this regime, however all the branches are asymptotically close, that is |q i (p) − q j (p)| 1 (1.18) for all branches q i , q j . Indeed, not only are the branches close, but the spectrum of the associated linearized operator is insensitive to the localized pulse positions i p ∈ Ktight , so that either Ktight ⊂ Kν for all branches of the tight regime, or i c Ktight ⊂ Kν for all branches q. We refer to the relative complement of the weak and tight regimes within Kl as the dynamic regime Kd . For pulse positions p ∈ Kd , 0 the spectrum changes by O(1) amounts as p varies across Kd . Assumption 1.1. We assume there exists a branch of N -pulse amplitude solutions i q i = q i (p) and a d0 > 0, independent of , and a nonempty open set K0 ⊂ Ki , which consists of pulse configurations, p, which are a minimum distance d0 from the boundary ∂K i of Ki . 14 i Specifically, we define K0 such that i K0 = p ∈ Ki | d p, ∂K i ≥ d0 , (1.19) i and introduce the adiabatic sub-manifold Mi to be the graph of Φp above K0 . When 0 i is fixed, we no longer notate it. Figure 1.4: This figure illustrates, for N = 2 and p1 < p2 , the minimum pulse separation l0 | ln | and also the pulse classes Ktight , Kd , and Kweak . To simplify our analysis, we impose the following restriction. 15 Simplification 1.1. We assume that α ≥ 0 and β ≥ 0 satisfy 1 − α/2 − β = 0. (1.20) This simplification assures that the delocalized component U (and Φ1 ) is O(1) in L∞ , which simplifies analysis. Coupled with (1.8), this limits β to 2/3 < β ≤ 1. The main result of this dissertation is an adiabatic stability result, which states that if the initial data to the system (1.4) begins sufficiently close to the adiabatic manifold M0 , then the full solution will decay to an asymptotically small layer of the adiabatic manifold M. There is also a time Tb , which is at least O −(1+ω) , for any 0 < ω < 1, and perhaps +∞, for which the full solution will remain inside this neighborhood of M, before p(t) hits ∂K, the boundary K. Once the full solution U has relaxed to the asymptotically small equilibrium layer of the adiabatic manifold M, we reduce the leading order pulse dynamics to an ordinary differential equation on the pulse positions p. At leading order, the evolution of the pulse positions with respect to time depends on the pulse positions p and the amplitude branch q = q(p) of the delocalized component at each pulse position. Assumption 1.1 affords the existence of a branch of adiabatic N -pulse solutions determined by q = q(p) over the domain K = Kl ,ν for some l0 , ν > 0 given. The 0 assumption also provides for a d0 > 0 such that K0 ⊂ K as defined in (1.19) is non-empty. 16 We introduce the spectral subset associated to the temporally decaying solutions ˜ of the semigroup generated by Lp : Xp = {U ||U ||X < ∞, πp U = 0}, (1.21) where πp is the spectral projection associated to the N -point spectrum σ0 near zero and the X-norm is defined in (1.40). More specifically, we state the adiabatic stability result and a leading order pulse dynamics result in the following theorem. Theorem 1.1. Adiabatic stability and leading order pulse dynamics Let > 0 be sufficiently small, while α and β satisfy Simplification 1.1 with α < 2/3. Fix ω ∈ (0, 1), then the adiabatic manifold of N -pulse solutions (1.12) afforded by Assumption 1.1 is adiabatically stable up to O (1+ω)(1−α/2) . That is, there exist M, M0 , Tb > 0 such that for all initial data U0 of (1.4) which lie within M0 α | ln |−2 of M0 in the X-norm (see (1.40)), the corresponding solutions of the system (1.4) can be uniquely decomposed as U (x, t) = Φp(t) (x) + W (x, t), (1.22) where Φp(t) is an adiabatic N -pulse solution and the remainder W ∈ Xp satisfies ||W (t)||X ≤ M α − 2 µt ||W0 ||X + (1+ω)(1−α/2) , e (1.23) for all 0 < t ≤ Tb −1−ω . Moreover, during this time interval the pulse dynamics 17 reduce to ∂p = 1+α/2 Qθ A(p)q −1 + O ∂t 2 , ||W (t)|| , ||W (t)||2 , X X (1.24) where Q is the diagonal matrix of pulse amplitudes q = q(p), the exponent is applied −1 T , and the antisymmetric matrix A(p) is −1 componentwise in q −1 = q1 , · · · , qN defined in (6.244). Elements of the Proof: A key construction of this work is the reduced linearization. The construction of the reduced linearization allows us to characterize the point spectrum as it evolves under the pulse evolution. The exact linearization of F about Φ(p) for p ∈ K takes the form  e −β V 11  L11 + Lp =  V21  −β V 12  , L22 (1.25) α α −1 2 2 where Le = −2 ∂x + α µ and L22 = ∂x − 1 + α22 Φ1 21 Φ2 22 . Also V11 11 and V12 are potentials described in detail in Chapter 4. The point spectrum for this linearization is not easily characterized. To understand the exact linearization and the reduced linearization, it is first useful and informative to examine their reductions. The spectra of the diagonal system   e  L11 0   , 0 L22 18 (1.26) is easy to characterize. The operator Le produces only essential spectra, while 11 σ (L22 ) consists of N positive ground eigenvalues, N eigenvalues clustered near zero, and the remainder of the spectrum strictly bounded on the negative real axis. This linearization generates an unstable flow and coupling is needed to generate stability. The spectrum is unchanged if we consider the lower-triangular system   e  L11 0   , V21 L22 (1.27) for any potential V21 . However, systems of the form  e −β J 11  L11 + ˜ Lp =  V21  −β J 12  , L22 (1.28) where J11 and J12 are finite rank operators, are sufficiently simple that their spectrum can be characterized, but are flexible enough to provide asymptotically accurate −1 approximations of the full system (1.25). Indeed, since Le is asymptotically 11 small away from long-wavelength functions, we show that for an appropriate choice of finite-rank operator J1i (i = 1, 2) the difference Le 11 −1 −β J1i − V1i is small as a map on the weighted-windowed space L1 discussed below. This differ1,p ˜ ence being small allows us to replace L with L without impacting the leading order dynamics. 19 ˜ The salient element of the reduction of L to L is to uniformly control the longwavelength elements. This is achieved through the weighted L1 norm, which 1,p N through a partition of unity χj centered about each pulse position, introduces j=1 locally weighted norms that control long-wavelength terms, uniformly for p ∈ K, in each χj window about the pulse at pj . Specifically, we define the L1 norm as 1,p N ||f || 1 = L 1,p j=1 1 + |x − pj | χj f 1 . L (1.29) Figure 1.5: This illustrates the second component pulse paired with the partition of unity χj at three pulse locations. Recalling the decomposition U = Φp(t) + W , we may rewrite the evolution equation (1.4) as an evolution for the remainder W and pulse positions p = p(t) (analyzed in Chapter 6), 20 ∂Φ ˙ ˜ Wt + p = R(Φ) + Lp W + ∆LW + N (W ), 0 ∂p (1.30) where the difference of the exact linearization and the reduced linearization is ˜ denoted ∆ = L − L and where W ∈ Xp , for the spectral subset Xp defined in (1.21). With the complimentary spectral projection πp , the W evolution is given by ˜ 0 Wt = −˜p π 0 ∂Φ ˙ ˜ p + Lp W + πp (∆LW + N (W )) . ˜ 0 0 ∂p (1.31) At an initial time tn , the renormalization group process freezes L = Lp , evolves n the fast system for a finite time, and then uses a non-linear solve to update the slow components in a self-consistent way. If the secularity in Lp − Lp can be controlled, n then uniform estimates are obtained on a finite time interval, and the process may be iterated. We introduce the renormalization times ti ∞ , where t0 is the initial i=1 time for (1.31). We introduce the quantity, α µ(s−ti ) ||W (s)||X , T1 (t) = sup e 2 ti 0 and the remainder can be appropriately bounded. Our analysis reduces the size of the remainder to a quadratic inequality, so either the remainder starts smaller than r1 and stays small or it begins larger the r2 . The middle interval (r1 , r2 ) is forbidden. It follows from the quadratic formula that for M0 < 1/(4C) and α < 2/3 there are two roots of g0 = 0 for 1. Moreover, with this bound on α we can take the renormalization group time period ti+1 − ti sufficiently long to obtain decay of W . Successive iterations start with a smaller bound on T1 (ti ) and yield tighter estimates on r1 , until a limit is reached and subsequent evolution yields no further decay. The renormalization group methodology yields the adiabatic stability result. The leading order pulse dynamics (evolution of p) are obtained by projecting equation (1.30) onto the tangent plane of M. After sufficient decay of the remainder W , the dominant term is given by the projection of F (Φ(p)), which yields (1.24). The lower bound on the time, Tb , is obtained as an upper bound on the pulse dynamics, with 22 M0 taken small enough that the remainder W decays to its adiabatic size before the pulses are within d0 /2 of ∂K. The following figure further illustrates the renormalization group process. Figure 1.7: This figure illustrates the renormalization group technique. The initial condition is decomposed as U0 = Φp0 + W0 (0), the linearization and associated spectral projections are frozen for a time interval sufficient to give decay, but not so long that the secular growth swamps the error. At the end of each renormalization interval, U (τ1 ) = Φ (p(τ1 )) + W (τ1 ) is reprojected into U (τ1 ) = Φ (p1 ) + W1 , where W1 ∈ Xp . The process is iterated and the transient associated to the ini1 tial perturbation decays to the level of the accuracy of the approximate adiabatic solution. 23 1.1 Notation We define the pulse positions p = (p1 , . . . , pN )T ∈ RN , (1.35) where N is the total number of pulses. We define the following norm: ||f || 1,1 = ||ξf || 1 + ||∂x f || 1 , L L W ξ (1.36) where ξ is a smooth, positive, compactly supported, mass one function, where ξj = ξ(x − pj ). The Sobolev-like norm W (1.37) 1,1 controls L∞ , since for any x, y ∈ R, ξ ∞ |u(x)| ≤ |u(y)| + −∞ |u (z)|dz, (1.38) x which uses the fact that u(x)−u(y) = y u dz. Multiplying by the mass-one function ξ(y), and integrating over all y ∈ R, we have ∞ |u(x)| ≤ ||u || 1 + |ξ(y)u(y)|dy = ||u|| 1,1 . L W −∞ ξ 24 (1.39) We define the following norm ||F ||X ≡ ||f1 || 1,1 + ||f2 || 1 , H W ξ (1.40) for F = (f1 , f2 )T . We define windowing a function f as N f= N fj = j=1 f χj . (1.41) j=1 N ×k N ×k For f ∈ L2 (R) and g ∈ L2 (R) , we define the tensor operator f ⊗ g, acting on h ∈ L2 (R) k by f ⊗ g · h = (h, g1 ) 2 f1 , . . . , h, gN 2 fN L L T ∈ L2 (R) N ×k , (1.42) where k denotes the number of components, so k = 2 for our system (1.4), and N is the number of pulses. 25 Chapter 2 Construction of an N -pulse Invariant Manifold. In this chapter, we construct an N -pulse invariant manifold for a reduction of the system (1.1). We fix N -pulse positions at p = (p1 , . . . , pN )T , and seek a manifold Φ = Φ(x; p) as a graph above an N -dimensional set p ∈ K ⊂ RN . More specifically, we seek Φ which satisfies the invariance condition: πT (p)F (Φ(p)) = 0, ˜ (2.1) where πT (p) is the projection complementary to the tangential plane of Φ, at Φ(p). ˜ The complementary tangential projection is written as πT ≡ I − πT , in terms of the ˜ 26 tangential projection: N πT f ≡ f, Bi i=1 B ∈ R2 . L2 i (2.2) N ×2 Here B = (B1 , . . . , BN )T ∈ L2 (R) is a Gram-Schmidt Orthonormalization N N ∂Φ ∂Φ of . We assume the family of vectors is linearly independent. ∂pi i=1 ∂pi i=1 So for each i, j: Bi , Bj 2 = δij , L (2.3) where δij is the Kronecker delta:     1, if i = j  . δij =   0, if i = j  (2.4) It is convenient to introduce the associated vector tangential projection: N πT f ≡ B ⊗ B · f = (f, B1 ) 2 B1 , . . . , (f, BN ) 2 BN L L T ∈ RN ×2 , (2.5) where B ∈ RN ×2 and f ∈ R2 . We show that if Φ satisfies (2.1) then its graph is invariant under the flow. For our general system (1.1), we decompose U as U = Φ(p) + W. 27 (2.6) Using Proposition 6.1, which allows us to determine a base point where πT W = 0, we choose W to be orthogonal to the tangent plane, so πT W = 0 and πT W = W . ˜ We linearize our system: ˙ p Φ · p + Wt = F(Φ) + LΦ W + N (W ). (2.7) Applying the projection and the complementary projection to (2.7), we have ˙ p Φ · p + πT Wt =πT F(Φ) + πT LΦ W + πT N (W ) (2.8) πT Wt =˜T F(Φ) + πT LΦ πT W + πT N (W ), ˜ π ˜ ˜ ˜ (2.9) where, by the construction of πT , ˙ pΦ · p = πT Indeed, we can represent each element of ∂Φ = ∂pi ˙ p Φ · p. (2.10) p Φ as N αk Bk , (2.11) k=1 so that ∂Φ πT = ∂pi N  N αk Bk , Bj  Bj =  j=1  N αj Bj = j=1 k=1 28 ∂Φ . ∂pi (2.12) Moreover since πT W = 0, it follows that 0= ∂Bi ∂ W, Bi 2 = Wt , Bi 2 + W, , L L ∂t ∂t L2 (2.13) from which we conclude N πT W t = − W, i=1 ∂Bi B. ∂t L2 i (2.14) Assuming the invariance condition (2.1) holds, we apply it to (2.9), which yields the flow N W, Wt + i=1 ∂Bi B = πT LΦ πT W + πT N (W ). ˜ ˜ ˜ ∂t L2 i (2.15) The set W = 0 is invariant under this flow. On the W = 0 manifold, we reduce to the tangential flow ˙ p Φ · p = πT F(Φ). (2.16) Thus a smooth solution Φ1 (p) of (2.1) yields an invariant manifold of the flow to (1.1), which reduces the flow to the ODE (2.16). Establishing a solution of (2.1) for our system (1.1) is beyond the scope of this 29 work. Instead we introduce the system:    F1 (U, V ; p, δ)  F(U ; p, δ) ≡  , F2 (U, V ; p, δ) (2.17) where  F1 ≡ −2 Uxx − α µU + −β (1 − δ) N  χj U (pj )α11 + δU α11  V α12 (2.18a) j=1  F2 ≡ Vxx − V − (1 − δ) K V 2 N + (1 − δ)  χj U (pj )α21 + δU α21  V α22 , j=1 (2.18b) with K(y) = √ −y ye , y ∈ [0, ∞) . (2.19) Here {χj }N is a partition of unity where each χj is a C ∞ function that is 1 on i=1 pj+1 −pj (sj−1 + 1, sj − 1) with sj = , for j = {1, . . . , N − 1} while s0 = −∞ 2 and sN = ∞. On (sj−1 − 1, sj−1 + 1] and [sj − 1, sj + 1), χj decays smoothly to zero. The V component is localized about the N -pulse positions p = (p1 , . . . , pN )T . The K(V ) term is added to remove any tail-tail interactions between these localized components. We consider δ ∈ [0, 1], where δ = 1 yields the original system (1.1), while δ = 0 30 Figure 2.1: Notice the function K(V / 2 ) exponentially decays for V >> 2 /2. reduces the inter-pulse coupling to a finite rank interaction. In particular F(U ; p, δ) = (1 − δ)F(U ; p, 0) + δF(U ), (2.20) where F(U ) is the unperturbed system that does not depend explicitly on p. In the remainder of this chapter, we will construct a manifold Φ(p) which satisfies F(Φ(p); p, δ = 0) = 0. (2.21) For δ > 0, we suggest how a contraction mapping argument could generate an invariant manifold with a slow normal velocity for δ = 1. We leave to posterity the 31 verification of this condition. However, after this chapter, we rigorously study the quasi-invariance of the δ = 0 manifold under the δ = 1 flow. 2.1 Explicit form for Φ2 We set δ = 0 in (2.17), and fix the pulse positions p = (p1 , . . . , pN )T . We look for Φ(x; p, 0) = (Φ1 , Φ2 )T which satisfies (2.21), that is, 0 = −2 ∂ Φ − α µΦ + −β xx 1 1 N α χj Φ1 (pj )α11 Φ2 12 j=1 N χj Φ1 (pj )α21 Φ2 α22 . 0 = ∂xx Φ2 − K(Φ2 / 2 ) − Φ2 + j=1 (2.22a) (2.22b) We introduce qj : qj ≡ Φ1 (pj ), (2.23) for j = 1, . . . , N . In this chapter, we solve the system q = q(p) = Φ1 (p). (2.24) We have the following theorem for the existence of this solution. Theorem 2.1. Self-consistent mean-field theorem Let δ = 0 for the system (2.18), Φ be defined in (2.44), and p ∈ K. Then there 32 exists an 0 > 0 such that for all satisfying 0 < Q(s, ) : RN × R → RN which is smooth in s and q = −κ Q η p, ≤ 0 , there exists a function such that , (2.25) where q, so defined, solves (2.72), and Φ = Φ (p, q(p)) solves (2.22). Here κ is defined as κ≡ 1 − α/2 − β θ−1 (2.26) for θ defined in (2.67), and η is defined as η ≡ 1 + α/2. Moreover, in the limit as (2.27) → 0, the rescaled variables q = κ q and p = η p solve ˜ ˜ q (˜) = M(˜)˜ θ , ˜p pq (2.28) where q θ = (˜1 , . . . , qN )T and M(˜) is defined in (2.82). ˜ qθ ˜θ p Furthermore, there exists constants k, K > 0 independent of and p ∈ K such that the scaled pulse amplitudes q are uniformly bounded above and below: ˜ k < ||˜|| < K. q 33 (2.29) Remark 2.1. After this Chapter, we assume (1.20), which in turn implies that κ = 0. Thus after Chapter 2, the unscaled pulse amplitudes q are equivalent to the scaled pulse amplitudes q , so by (2.29), the unscaled pulse amplitudes q are bounded ˜ above and below. We construct our ansatz where we show (2.22b) has an exact solution of the form Φ2 (x) = N φ (x), where each φ is compactly supported on (s j j−1 + 1, sj − 1) j=1 j and φj satisfies α 0 = φj − K(φj / 2 ) − φj + qj 21 φj α22 . (2.30) This equation has a first integral φj2 = H(φj ) + 2 where H(x) = α qj 21 φ2 j − φj α22 +1 , 2 α22 + 1 (2.31) x 2 0 K(s/ )ds. Isolating the left side, we define the homoclinic solution φj by the initial value problem      φ = Ω(φj ) = ± j   φ (p ) =  j j φ∗ j  α21   2q  α22 +1  2− j 2H(φj ) + φj α +1 φj , 22     where Ω(φ∗ ) = 0. j 34 (2.32) Figure 2.2: An illustration of the homoclinic orbit of φj . For |φj | 2 , we consider an asymptotic expansion of φ , j φj = φ0 + O( ). j (2.33) Plugging this expansion into (2.30), at leading order we have α α 0 = φ0 − φ0 + qj 21 φ0 22 . j j j (2.34) We can solve this exactly on (sj−1 + 1, sj − 1): φ0 (x − pj ) φ0 (x) = , j α /(α22 −1) qj 21 35 (2.35) where φ0 (x − pj ) is a pulse centered at x = pj . We have that φ0 (x) << 2 well j before x nears the boundaries of (sj−1 + 1, sj − 1) as long as the pulse separation is pj − pj−1 O(| ln |). The set Kl where this condition is met is defined in (1.9). 0 Since φj is homoclinic to zero, we know that φj → 0 as |x − pj | → ∞. Once φj 2 , we can expand the exponential part of K about zero:  3/2 φj   . φj + O  2   K φj / 2 = (2.36) In this regime, (2.22b) will asymptotically scale to 0 = φj − φj . (2.37) If we multiply by φj and integrate, we have φj = − 4 3/4 φ . 3 j (2.38) Separating variables, integrating, and simplifying, leads to φj (x) = 1/144 (x − c)4 . (2.39) The above is the tail behavior of φj , giving it compact support on the pulse interval 36 (sj−1 , sj ). When φj (x) = O( 2 ), (x − c) = O( 1/2 ), where c can be determined to appropriately match function values. We define Φ2 to be the sum of pulses that meets these asymptotic conditions: N Φ2 ≡ φj (x). (2.40) j=1 Then for any p ∈ Kl and q(p) ∈ RN , Φ2 is a steady state F(Φ2 ; p, δ = 0) = 0. 0 Figure 2.3: This is a cartoon of the pulse shape of Φ2 at the pulse position pj . Notice how it is localized. 37 2.2 Explicit form for Φ1 To determine Φ1 , we consider (2.22a) with our solution Φ2 : Le Φ1 = −β 11 = −β N j=1 N α χj qj 11 Φ2 α12 (2.41) α α qj 11 φj 12 (x), (2.42) j=1 where 2 Le = − −2 ∂x + α µ. 11 (2.43) The inverse of Le is denoted L−e . We define Φ1 as the solution to (2.42). In 11 11 summary, we define Φ to be    −β L−e  Φ1 (x, p, q)   11 Φ(x, p, q) ≡  = Φ2 (x, p, q) subject to (2.23). 38 N q α11 φ α12 (x) j j=1 j N φ (x) j=1 j   , (2.44) 2.3 Construction of the pulse amplitudes q(p) The condition qj = Φ(pj ), requires q to satisfy the system  qk = −β L−e  11  N α qj 11 φj α12 (pk ) , (2.45) j=1 for k = 1, . . . , N. The Green’s function Gλ (x) associated to Le + λ has the property 11 that Le + λ −1 f = Gλ ∗ f (x). 11 (2.46) Using the Fourier transformation, Gλ is found to be Gλ (x) ≡ π 2 −k |x| e λ , 2 kλ (2.47) where we introduce, kλ ≡ λ + α µ. (2.48) For λ = 0, we have G0 (x) = π 1−α/2 − 1+α/2 √µ|x| e . 2µ 39 (2.49) We proceed with an asymptotic reduction of (2.45). Applying (2.46) to (2.45) with λ = 0, we have  qk = −β G0 ∗ N  α qj 11 φj α12  (pk ). (2.50) j=1 Writing this in vector form leads to: q = 1−α/2−β Gq α11 , (2.51) α α with q α11 = (q1 11 , . . . , qN11 )T and G ∈ RN ×N , where α Gj,k = α/2−1 G0 ∗ φj 12 (pk ), (2.52) so G has no leading power in . We asymptotically expand G by substituting our asymptotic expansion (2.33) for φj . Here ˜ φj = φ0 + φj , j (2.53) ˜ where φj has compact support and both φ0 and φj decay exponentially. We define j ˜ G = G 0 + G, 40 (2.54) where α 0 Gj,k = α/2−1 G0 ∗ φ0 12 (pk ), j (2.55) and from a Taylor expansion and (2.35), we determine α −1 ˜ ˜ φj Gj,k =α12 α/2−1 G0 ∗ φ0 12 j (pk ) + O( ) (2.56) α −1 ˜ −α (α −1)/(α22 −1) =α12 α/2−1 qj 21 12 G0 ∗ φ0 12 φj j (pk ) + O( ). (2.57) α −1 via (2.35). We The extra factors of qj above are a result of replacing φ0 12 j write this matrix as: ˜ ˜ G = G red Q −α21 (α12 −1)/(α22 −1) , with the N × N diagonal matrix Qij = qi , i = j 0, i = j (2.58) and ˜ ˜red Gj,k = α12 α/2−1 G0 ∗ φ0 α12 −1 φj (pk ). 41 (2.59) Substituting G = G 0 into (2.51), we have, for any k,  qk = −β G0 ∗  N α qj α11 φ0 12  (pk ) j (2.60) j=1 = = N √ α α − 1+α/2 µ|pk −y| e qj 11 φ0 12 (y)dy j j=1 N 1+α/2 √µ|p −y| α11 0 α12 π 1−α/2−β k (y)dy. e− qj φj 2µ j=1 π 1−α/2−β 2µ (2.61) (2.62) 1+α/2 √µ|p −y| k Next, we Taylor expand e− about pj : e− 1+ α √ 2 µ|pk −y| =e− − 1+ α √ 2 1+ α √ 2 µ|pk −pj | µe− (2.63) 1+ α √ 2 µ|pk −p∗ | (y − pj ) + O( 2+α ). (2.64) Substituting this into the integral in (2.62) and recalling (2.35), we have that: √ 1+α/2 √µ|p −p | − 1+α/2 µ|pk −y| q α11 φ0 α12 (y)dy =φα12 q θ e− k j e j j 0 j θ +h( , φ0 )qj , (2.65) ∞ where we define the mass f = −∞ f (s)ds. We also have h( , φ0 ) = O( 1+α/2 ), 42 (2.66) since the decay from φ0 dominates the polynomial terms from the Taylor expansion. j We define θ ≡ α11 − α12 α21 /(α22 − 1). (2.67) Then with the previous reductions, at leading order q = 1−α/2−β M(p, )|q |θ , (2.68) T where |q |θ ≡ |q1 |θ , . . . , |qN |θ . The matrix M is defined as α Mj,k (p, ) ≡ φ0 12 α/2−1 GN = ij √ π α12 − 1+α/2 µ|pk −pj | φ e , 2µ 0 (2.69) where GN = GN (p) is defined as the two-point correlation matrix: 0 0 [GN ] ≡ GN = 0 ij √ π 1−α/2 − 1+α/2 µ|pi −pj | e . 2µ (2.70) We include the absolute value in (2.68) because we want to show the existence of a nontrivial q with every component positive. If we return to our exact equation (2.51), and substitute (2.54) for G, we have ˜ q = 1−α/2−β M(p, )|q |θ + 2−α/2−β G|q |α11 + O( 1+α/2 )|q |θ . 43 (2.71) ˜ To remove any q dependence in G, we recall (2.58), and rewrite the above as α θ+ α 21 ˜ 22 −1 + O( 1+α/2 )|q |θ . q(p) = 1−α/2−β M(p, )|q |θ + 2−α/2−β G red |q | (2.72) To eliminate the 1−α/2−β from the leading term above, we rescale the amplitude variable as q = κ q. ˜ (2.73) 1 − α/2 − β , θ−1 (2.74) We have κ≡ where (2.72) becomes: α21 θ + ρ G red (p, )|˜|θ+ α22 −1 + O( α+β )|˜|θ , ˜ q q ≡ M(p, )|˜| ˜ q q (2.75) with ρ=1+ α21 α22 − 1 1 − α/2 − β 1−θ . (2.76) We need the condition that ρ > 0, 44 (2.77) to analyze the appropriate leading order problem. We rescale the pulse position variable to remove any ˜ dependence from M(p, ) (and also from Gred ): p = η p, ˜ (2.78) η = 1 + α/2. (2.79) q = A(˜, q , ), ˜ p ˜ (2.80) with With this rescaling, (2.75) becomes where α21 θ + ρ G red (˜)|˜|θ+ α22 −1 + O( α+β )|˜|θ , ˜ q A(˜, q , ) ≡ M(˜)|˜| p ˜ p q p q (2.81) and, in the rescaled variable, M(˜) has the componentwise form p Mj,k (˜) ≡ p √ π α12 − µ|˜k −˜j | p p . φ0 e 2µ 45 (2.82) First we examine the = 0 case. The system (2.45), under this rescaling, reduces to q = A(˜, q , 0) = M(˜)|˜|θ . ˜ p ˜ p q (2.83) If θ = 0, we have the nontrivial positive solution where for each i: N qi = ˜ Mi,j . (2.84) j=1 We consider the case when θ = 0 and θ = 1. Examining (2.82) and (2.83), it is clear that Ai (˜, q , 0) = M(˜)|˜|θ ≥ 0 p ˜ p q (2.85) for all q ∈ RN and each i ∈ (1, . . . , N ). Indeed, from (2.82) we see that every ˜ entry of M is positive and uniformly bounded. This also implies that there exists 0 < k < K, which depend upon p, such that ˜ k||˜||θ ≤ ||A(˜, q , 0)|| ≤ K||˜||θ q p ˜ q (2.86) for some vector norm || · ||. We define B(˜, q , 0) ≡ p ˜ 46 A(˜, q , 0) p ˜ , ||˜||θ q (2.87) and S ≡ {˜ : k ≤ ||˜|| ≤ K, qi ≥ 0, i ∈ (1, . . . , N )}, q q ˜ (2.88) where q = q1 , . . . , qN T . S is compact and B maps S into itself. S is a contractible ˜ ˜ ˜ Figure 2.4: This is an illustration of S in two space dimensions. manifold, where a contractible manifold is defined to be a manifold that can be continuously shrunk to any point inside itself. Moreover, B is continuous on S. Then by the Eilenberg-Montgomery fixed point theorem (which is a corollary to the Brouwer fixed point theorem), B has a fixed point q∗ ∈ S. ˜ We have that A(˜, q∗ , 0) = ||˜∗ ||θ q∗ , p ˜ q ˜ 47 (2.89) and similarly for any constant s we have that A(˜, s˜∗ , 0) = sθ A(˜, q∗ , 0) = sθ−1 ||˜∗ ||θ s˜∗ . p q p ˜ q q (2.90) s ≡ ||˜∗ ||−θ/(θ−1) , q (2.91) A(˜, q∗ , 0) = q∗ , p (2.92) q∗ ≡ s˜∗ . q (2.93) We choose so that A has the solution: where So there exists a nontrivial solution to (2.83) with every component positive. To show that solutions to (2.80) are locally unique for > 0, we need the non-bifurcation condition (2.96) that gives generic local continuation. For p ∈ Kν , we have a locally unique solution to (2.80). Doing a continuation argument using the implicit function theorem, we define B(˜, q , ) = q − A(˜, q , ). p ˜ ˜ p ˜ (2.94) We have from (2.92) that B(˜, q∗ , 0) = 0. Differentiating B(˜, q , ) with respect to q , p p ˜ ˜ 48 evaluated at (˜, ) = (q∗ , 0): q q B(˜, q∗ , 0) = I − ˜ p p p θ−1 , q A(˜, q∗ , 0) = I − θM(˜)|q∗ (˜)| ˜ p (2.95) where M(˜) was defined at (2.82). Then the non-bifurcation condition is p det I − θM(˜)|q∗ (˜)|θ−1 = 0. p p (2.96) The set Kν is the maximal open set of all pulse positions p beginning in the wellseparated regime, and continuously extended into the semi-strong regime until values of p where (2.96) fails. By the implicit function theorem, for all p = −η p ∈ Kν , ˜ there exists 0 > 0 such that for 0 ≥ > 0, we may extend: q∗ = q∗ (˜, ), ˜ ˜ p (2.97) which solves (2.81) smoothly in p and . Then in the original variables p and q, (2.97) ˜ is equivalent to (2.25) in Theorem 2.1, listed again as q = −κ Q η p, , where Q is equivalent to the continuous extension q∗ in (2.97). ˜ 49 (2.98) 2.4 Existence of an invariant manifold We continue examining the case δ = 0. From our positive solution q∗ to (2.83), we ˜ p ˜ p ˜ construct Φ∗ (˜). We have F(Φ∗ (˜); p, δ = 0) = 0, so ˜ p ˜ πT F(Φ∗ (˜); p, δ = 0) = 0. ˜ (2.99) Thus we have an invariant manifold for the case δ = 0. This case is even stronger in that we have a steady state solution, where the N -pulse positions are fixed. For the case when δ ∈ (0, 1], we want to preserve invariance, so we need (2.1) to hold. By continuation, we will show a condition that will ensure πT (Φ(p, δ), p) (F (Φ(p, δ); p, δ)) = 0, ˜ (2.100) where πT (Φ(p, δ), p) depends on Φ(p, δ) and p. We differentiate the above with ˜ respect to δ: ∂Φ ˜ Φ πT ∂δ ∂Φ ∂F (F (Φ(p, δ); p, δ)) + πT (Φ(p, δ), p) Lp,δ ˜ + (Φ(p, δ); p, δ) ∂δ ∂δ = 0. (2.101) At δ = 0, the first term is zero, so πT (Φ(p, 0), p) Lp,0 ˜ ∂Φ(p, 0) ∂δ = −˜T (Φ(p, 0), p) π 50 ∂F (Φ(p, 0); p, 0) . ∂δ (2.102) This is equivalent to −1 ∂Φ(p, 0) = − πT Lp,0 πT ˜ ˜ ∂δ ∂F (Φ(p, 0); p, 0) . ∂δ (2.103) If (2.103) holds, we have the existence of an invariant manifold. A key step is to characterize the bounded invertibility of the conjugated operator πT Lp,0 πT . ˜ ˜ Instead, we will continue our analysis of the case when δ = 1 by using an approximate solution Φ. This will lead to a quasi-invariant manifold using a spectral projection defined after we analyze our eigenvalue problem. 51 2.5 Numerical Results This section contains two numerical results. The first examines (2.22b) with the values Φ1 (pj )α21 = 1 and α22 = 2. We further simplify the system by examining only one pulse position and replacing Φ2 with Y . This choice of variables leads to: Yxx = K(Y / 2 ) + Y − Y 2 . (2.104) Our boundary conditions are that Y (0) = 0 and Y (10) = 0. We use the fact that 3 Y = 2 sech2 (Y /2) is a solution to this equation when the K term is not present to construct a solution to this equation with the K term present. Using the Matlab boundary value solver bvp4c, and solving for x ∈ [0, 10], we have the following results in Figure 2.5 and Figure 2.6. Our second numerical results demonstrates the behavior of the pulses and amplitudes in our system. Given pulse positions p, we solve the nonlinear system (2.83) for the amplitudes q. The system we are solving is q = M(p)|q|θ , for M(p) defined in (2.69), and θ applied componentwise. Reducing the problem to three pulses, we take = 0.1, α = 0, and the coefficients of M to be 1. We solve this system beginning in the well-separated pulse regime, where M is almost diagonal, and continue the pulse positions closer together. As the Figure 2.7 illustrates, for θ = −1, the pulse amplitudes converge as the pulse positions converge. 52 Figure 2.5: This figure is a graph of the solution to (2.104) without the K term in blue and the full solution to (2.104) in red. The solution to the full equation decays faster. Figure 2.6: The fact that the solution to the full equation decays faster is more evident in this semi-log plot in the vertical coordinate. Again, the solution to (2.104) absent the K term is in blue and the solution to the full equation (2.104) is in red. 53 Figure 2.7: Each color is one component of the pulse amplitude, where pulse separation is varied which results in a change in amplitude. The horizontal axis is a log-scale of the separation between pulses, continued from a well-separated regime to a semi-strong regime. The values given are |p2 − p1 |. The vertical axis is pulse amplitude. 54 Figure 2.8: These two graphs illustrate the three pulses and their amplitudes for |p2 − p1 | = 33 on the left and |p2 − p1 | = 1.6 on the right, corresponding to the previous figure. 55 Chapter 3 Original System: Ansatz and Residual Estimates We return to the system (1.1), ∂t U = ∂t V −2 U xx − α µU + −β U α11 V α12 = Vxx − V + U α21 V α22 . (3.1a) (3.1b) We show that the invariant manifold we constructed in Chapter 2 for the reduced system is a sufficiently accurate approximate invariant manifold for the original system to capture the leading order dynamics. We construct our ansatz as in the last chapter, where we solve (3.1b) for V = Φ2 at equilibrium, with U = Φ1 approxi56 mated by qj = Φ1 (pj ) at each pulse position. We then solve (3.1a) at equilibrium for q = Φ1 (p). This amounts to solving (2.22) with the − K(Φ2 / 2 ) term dropped. With this slightly modified construction, we define    −β L−e  Φ1   11 Φ≡ ≡ Φ2 N q α11 φα12 (x) j=1 j j N φ (x) j=1 j   , (3.2) where φj solves α α 0 = φj − φj + qj 21 φj 22 . (3.3) It is also defined to be φj (x) ≡ φ0 (x − pj ) , α21 /(α22 −1) qj (3.4) where φ0 (x) solves α φ0 − φ0 + φ0 22 = 0. 3.1 (3.5) Linear estimates The following lemma contains estimates used to develop resolvent estimates on the linearization, and thus to derive our semigroup estimates. The following estimates are used to obtain estimates on Φ1 , which lead to residual estimates. Recall the 2 definition of Le = − −2 ∂x + α µ from (2.43), kλ ≡ 11 57 λ + α µ defined in (2.48), and that the χj ’s form a partition of unity about the pulse positions p. We have the following lemma. 1,1 Lemma 3.1. There exists C < ∞ such that for any f ∈ L1 (R) or f ∈ W (R) ξ and λ ∈ C \ (−∞, − α µ), the following estimates hold: 2 ||(Le + λ)−1 f || 1,1 ≤ C ||f || 1 , 11 L Re(kλ ) W ξ 2 ||(Le + λ)−1 f || 1,1 ≤ C ||f || 1,1 , 11 |kλ |Re(kλ ) W W ξ ξ 2 ||f || 1 , ||(Le + λ)−1 f ||L∞ ≤ C 11 L |kλ | ||∂x ((Le + λ)−1 f )||L∞ ≤ C 2 ||f || 1 , 11 L 2 ||f || 1 , ||∂x ((Le + λ)−1 f )|| 2 ≤ C 11 L L Re(kλ )1/2 2 ||∂x ((Le + λ)−1 f )|| 1 ≤ C ||f || 1 , 11 L L Re(kλ ) 2 ||f || 1 , ||(Le + λ)−1 f || 1 ≤ C 11 L L |kλ |Re(kλ ) 2 ||f || 2 . ||(Le + λ)−1 f || 2 ≤ C 11 L L |kλ |Re(kλ ) 58 (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) Moreover, for any f ∈ L1 (R), we have the following estimates 1,p 2 e + λ)−1 f || ||(L11 1,1 ≤ C Re(k ) W λ ξ ||(Le + λ)−1 f ||L∞ ≤ C 2 11 | ⊗ χ · f | + kλ ||f || 1 L 1,p 1 | ⊗ χ · f | + ||f || 1 L |kλ | 1,p , . (3.14) (3.15) Proof: We define g(x) ≡ (Le + λ)−1 f = (Gλ ∗ f )(x). Recall that Gλ (x) = 11 π 2 −k |x| e λ . From the identity g = Gλ ∗ f and the Lp convolution estimates 2 kλ [25], we have 2 ||g || 1 ≤ ||Gλ || 1 ||f || 1 ≤ C ||f || 1 . (3.16) L L L L Re(kλ ) Similarly, 2 ||f || 1 . ||ξg|| 1 ≤ ||ξ|| 1 ||(Gλ ∗ f )(x)||L∞ ≤ ||Gλ ||L∞ ||f || 1 ≤ C L L L L |kλ | (3.17) These two estimates establish (3.6). To prove (3.8) we observe that 2 ||f || 1 . ||g||L∞ = ||(Gλ ∗ f )(x)||L∞ ≤ ||Gλ ||L∞ ||f || 1 ≤ C L L |kλ | (3.18) We have (3.9), since ||g ||L∞ = ||(Gλ ∗ f )(x)||L∞ ≤ ||Gλ ||L∞ ||f || 1 ≤ C 2 ||f || 1 . L L 59 (3.19) In a similar manner we obtain (3.10), since ||Gλ || 2 = C L 4 e−2kλ |x| dx1/2 ≤ C 2 Re(kλ )1/2 . (3.20) The next inequality (3.11) follows similarly from the estimate 2 2 e−kλ |x| dx ≤ C . ||Gλ || 1 = C L Re(kλ ) R (3.21) For (3.12), 2 ||g|| 1 = ||(Gλ ∗ f )(x)|| 1 ≤ ||Gλ || 1 ||f || 1 ≤ C ||f || 1 . L L L L L |kλ |Re(kλ ) (3.22) Similar to this, for (3.13), 2 ||f || 2 . ||g|| 2 = ||(Gλ ∗ f )(x)|| 2 ≤ ||Gλ || 1 ||f || 2 ≤ C L L L L L |kλ |Re(kλ ) (3.23) Next we prove (3.7). For this case, we first observe that ||ξg|| 1 ≤||ξ|| 1 ||(Gλ ∗ f )(x)||L∞ L L ≤||Gλ || 1 ||f ||L∞ L 2 ≤C ||f || 1,1 . |kλ |Re(kλ ) W ξ 60 (3.24) (3.25) (3.26) We have our result if we combine the previous line with 2 ||g || 1 = ||(Gλ ∗ f )(x)|| 1 ≤ ||Gλ || 1 ||f || 1 ≤ C ||f || 1,1 . (3.27) L L L L |kλ |Re(kλ ) W ξ For the final two estimates, we decompose f as in (1.41), where f = j fj and fj = χj f , so that gj = Gλ ∗ fj satisfies g = (3.28) j gj . Moreover using (1.29), we see that ||fj || 1 , ||f || 1 = L1,j L 1,p j (3.29) where || · || 1 = || 1 + |x − pj | · || 1 . We proceed by decomposing each fj into L L1,j a small mass and a massless part: ¯ fj = fj ξj + yj , (3.30) for yj ∈ L1 (R) and ξj defined in (1.37). Clearly for any f , ||f || 1 ≤ ||f || 1 . Next L L1,j 61 we examine ||yj || 1 = L R ≤ R ∂x (x − pj ) |yj |dx |(x − pj )yj |dx =C R ≤C (3.31) (3.32) ¯ |(x − pj ) fj − fj ξj |dx ||fj || 1 + ||f || 1 |(x − pj )ξj |dx L R L1,j ≤C||fj || 1 . L1,j (3.33) (3.34) (3.35) ¯ We decompose gj = gj,1 +gj,0 where gj,1 = fj Gλ ∗ξj and gj,0 = Gλ ∗yj = Gλ ∗yj . Estimating gj,1 using (3.6), we have 2 2 ¯ ¯ ¯ ||gj,1 || 1,1 = fj ||Gλ ∗ ξj || 1,1 ≤ C fj ||ξj || 1 ≤ C f . L Re(kλ ) Re(kλ ) j W W ξ ξ (3.36) The function Gλ has a jump at x = 0. We deduce that ∂x gj,0 = 2Gλ ∗ yj = Gλ ∗ yj + 2 yj , (3.37) so that      x=0  Gλ Gλ = ,  2   δ  x=0 x = 0 62 (3.38) where the function Gλ is the point-wise second derivative of Gλ . Then |kλ | 2 ||gj,0 || 1 ≤ || Gλ || 1 + 2 ||yj || 1 ≤ C ||fj || 1 . L L L L1,j Re(kλ ) (3.39) Using (3.9), ||ξgj,0 || 1 ≤ ||ξ|| 1 ||Gλ ∗ yj ||L∞ ≤ C||Gλ || 1 ||yj || 1 ≤ C 2 ||fj || 1 . (3.40) L L L L L1,j Summing over j, we have (3.14). The final inequality (3.15) follows using (3.9) and (3.8) respectively: ||gj,0 ||L∞ =||Gλ ∗ yj ||L∞ ≤ C||Gλ ||L∞ ||yj || 1 ≤ C 2 ||fj || 1 L L1,j 2 ¯ ¯ ¯ f ||ξgj,0 ||L∞ =fj ||Gλ ∗ ξj ||L∞ ≤ C fj ||Gλ ||L∞ ||ξj || 1 ≤ C L |kλ| j (3.41) (3.42) Recall from the introduction the assumption that 1 − α/2 − β = 0. We use this assumption and the previous lemma to prove the following bounds on Φ1 (p, δ = 1). In particular, the lemma assures that Φ1 = O(1) in L∞ . 63 Lemma 3.2. There exists a constant C < ∞ such that ||Φ1 ||L∞ ≤C (3.43) ||∂x Φ1 ||L∞ ≤C 2−β (3.44) ||Φ1 || 1 ≤C β−2 L ∂qj ||L∞ ≤C 2−β || ∂pk (3.45) (3.46) ||∂p Φ1 || 1 ≤C k L (3.47) ||∂pk Φ1 ||L∞ ≤C 2−β . (3.48) Proof: For the first inequality we use (3.8) and the assumption 1 − α/2 − β = 0:  ||Φ1 ||L∞ = −β L−e  11 N  α α qj 11 φj 12  L∞ j=1 N ≤C α α qj 11 φj 12 j=1 (3.49) (3.50) L1 ≤ C. (3.51) 64 For (3.44) we have  ||∂x Φ1 ||L∞ ≤  N C −β ∂x L−e  11  α α qj 11 φj 12  L∞ j=1 ≤ C 2−β N α α qj 11 φj 12 j=1 ≤ (3.52) (3.53) L1 C 2−β , (3.54) where we used (3.9). For (3.45), using (3.12) we have  ||Φ1 || 1 ≤ L ≤  C −β L−e  11 N  α α qj 11 φj 12  j=1 C β−2 N L1 α α qj 11 φj 12 j=1 ≤ (3.55) (3.56) L1 C β−2 . (3.57) For (3.46), we examine qj from the exact formulation (2.72) at leading order, || ∂ M(p, )|q |θ ∂qj || ∞ ≤C ∂pk L ≤C ∂pk ∂M(p, ) ∂pk L∞ ≤C 2−β , 65 j (3.58) L∞ (3.59) (3.60) due to the α/2+1 = 2−β in the exponent of M(p, ) defined in (2.69). For (3.47), we have that  ∂pk Φ1 = −β L−e ∂pk  11 N  α α qj 11 φj 12  (3.61) j=1  α α = −β L−e −q 11 ∂x φ 12 − 11 k k N θ−1 α θqj φ0 12 j=1 ∂qj ∂pk  . (3.62) Then applying the L1 norm to the above α ||∂pj Φ1 || 1 ≤ −β C||Gλ ∗ ∂x φ 12 || 1 + −β C||L−e 11 k L L N θ−1 α ∂qj || . θqj φ0 12 ∂pk L1 j=1 (3.63) For the first term on the right, −β C||G ∗ ∂ φα12 || =||∂x Gλ ∗ x k λ L1 −β q α11 φα12 k k α α ≤C β || −β q 11 φ 12 || 1 k k L ≤C, || 1 L (3.64) (3.65) (3.66) where we applied the estimate (3.11). For the second term on the right in (3.63), 66 applying the estimate (3.12) and (3.46) we have −β C||L−e 11 N N θ−1 φα12 ∂qj || θ−1 α ∂qj || −β C θqj ≤ ||θqj φ0 12 0 ∂p L1 ∂pk L1 k j=1 j=1 N ∂qj α β−2 C ≤ || ∞ ||φ0 12 || 1 || L ∂pk L j=1 (3.67) (3.68) ≤C. (3.69) ||∂p Φ1 || 1 ≤ C. k L (3.70) We conclude that Proving (3.48) follows similarly where α ||∂pj Φ1 ||L∞ ≤ −β C||Gλ ∗ ∂x φ 12 ||L∞ k N θ−1 α ∂qj || −β C||L−e + θqj φ0 12 ∞. 11 ∂pk L j=1 (3.71) The first term on the right follows like above, using (3.9), −β C||G ∗ ∂ φα12 || ∞ =||∂ G ∗ x k x λ λ L −β q α11 φα12 k k α α ≤C 2 || −β q 11 φ 12 || 1 k k L ≤C 2−β . 67 ||L∞ (3.72) (3.73) (3.74) For the second term on the right in (3.71), applying the estimate (3.8) and (3.46) we have −β C||L−e 11 N θ−1 α ∂qj || θ−1 φα12 ∂qj || ||θqj φ0 12 θqj ∞ ≤C 0 ∂p L ∂pk L1 k j=1 j=1 N ∂qj α ≤C || ∞ ||φ0 12 || 1 || L ∂pk L j=1 N (3.75) (3.76) ≤C 2−β . (3.77) ||∂p Φ1 ||L∞ ≤ C 2−β k (3.78) Then we conclude From the proof of (3.47), we conclude that ∂φj ∂Φ2 +O =− ∂pj ∂x 2−β , (3.79) in any Lp norm. The following corollary will later be used to determine the point spectrum. Corollary 3.1. There exists C > 0 such that for all λ ∈ C \ (−∞, − α µ) and p ∈ K, the following holds (Le + λ)−1 f, g 2 − (⊗χ · f )T GN ⊗ χ · g ≤ C 2 ||f || 1 ||g|| 1 , 11 λ L L L 1,p 1,p 68 (3.80) for all f, g ∈ L1 , where L1 is defined in (1.29) and GN (for λ = 0) is the 0 1,p 1,p two-point correlation matrix defined in (2.70). Proof: To show (3.80) we use the Taylor expansion Gλ (y − x) =Gλ pi − pj + (y − pi ) − (x − pj ) =Gλ (pi − pj ) + Gλ (s) (y − pi ) − (x − pj ) , (3.81) (3.82) for some s ∈ R. Windowing f and g as in (1.41) and substituting the above we have: (Le + λ)−1 f, g 2 = 11 L N (Le + λ)−1 fi , gj 11 i,j=1 N L2 Gλ (y − x)fi (y)gj (x)dydx = (3.83) (3.84) i,j=1 N Gλ (pi − pj )fi (y)gj (x)dydx = i,j=1 N Gλ (s) (y − pi ) − (x − pj ) fi (y)gj (x)dydx + i,j=1 (3.85) =(⊗χ · f )T GN ⊗ χ · g λ N (y − pi ) − (x − pj ) fi (y)gj (x)dydx. +Gλ (s) i,j=1 (3.86) 69 We rearrange the above terms and apply the absolute value: (Le + λ)−1 f, g 11 − (⊗χ · f )T GN ⊗ χ · g = |Υ|, λ (3.87) (y − pi ) − (x − pj ) fi (y)gj (x)dydx. (3.88) L2 where N Υ = Gλ (s) i,j=1 Using the fact that ||Gλ ||L∞ ≤ C 2 , we estimate the last term of (3.87), N |Υ| = Gλ (s) ≤C 2 ≤C 2 (y − pi ) − (x − pj ) fi (y)gj (x)dydx (3.89) i,j=1 N |y − pi | + |x − pj | |fi (y)gj (x)|dydx (3.90) i,j=1 N |(y − pi )fi (y)|dy |gj (x)|dx + |(x − pj )gj (x)|dx |fi (y)|dy i,j=1 (3.91) ≤C 2 ||f || 1 ||g|| 1 L L 1,p 1,p (3.92) 70 3.2 Residual estimates The residual is R(Φ) = F(Φ), which takes the form  −2 ∂ 2 Φ − α µΦ + −β Φα11 Φα12 x 1 1   R1 (Φ)   2 1 R(Φ) =  . = α21 α22 2Φ − Φ + Φ ∂x 2 R2 (Φ) 2 1 Φ2    (3.93) We have the following properties for the residual: Proposition 3.1. Recall the definition of Kl in (1.9). Fix l0 from this definition, 0 then for all p ∈ Kl , the residual has the following asymptotic formula 0     R1 (Φ)    = R2 (Φ)  N (Φα11 − q α11 )φα12 + O( r )  j=1 1 j j , α21 α21 α22 N r) φj + O( j=1 Φ1 − qj −β for r = r(l0 ) > 0 large. Moreover, there exists C > 0, independent of (3.94) and p ∈ Kl 0 such that for all p ∈ Kl the following estimates hold, 0 ||R1 (Φ)|| 1 ≤ C α L (3.95) ||R2 (Φ)|| 2 ≤ C 2−β . L (3.96) Proof: We first examine R2 (Φ) in the L2 norm. Adding and subtracting like 71 terms, we find N ||R2 (Φ)|| 2 ≤ L || j=1 α α 2 ∂x φj − φj + qj 21 φj 22 || 2 L  α Φ1 21  + N α φj  22 N − j=1 N || + j=1  α φj 22  j=1 L2 α α α Φ1 21 − qj 21 φj 22 || 2 . L (3.97) The first term above is zero by the definition of φj from (3.3). Next, using (3.43):  α 22 N α α ≤ φj 22  Φ1 21  φj  − j=1 j=1 L2 ≤  since N α C r ||Φ1 21 ||L∞ (3.98) C r, (3.99) α 22 N α  φj  − φj 22 ≤ C r, j=1 j=1 L2  N (3.100) for r ≥ 2, which follows from the fact that p ∈ Kl . In this space, the pulses are 0 sufficiently separated so that the tail-tail interaction between φj and φk for j = k α is minimal. Finally, for the third term we Taylor expand Φ1 21 under the sum at 72 x = pj for each j and use (3.44): N N α21 α21 α22 α α || (Φ1 − qj )φj || 2 ≤ || ∂x Φ1 21 (sj )|x − pj |φj 22 || 2 (3.101) L L j=1 j=1 N α α ≤ |∂x (Φ1 21 )(sj )| |x − pj |φj 22 2 (3.102) L j=1 N α ≤C (3.103) ||∂x (Φ1 21 )||L∞ j=1 ≤ C 2−β . (3.104) For the above, we used (3.43) and (3.44), where sj ∈ R for each j, and the exponential α decay in φj 22 dominates the linear growth of |x − pj |. From (3.97), (3.99), and (3.104), we conclude that ||R2 (Φ)|| 2 ≤ C 2−β , L (3.105) which establishes (3.96). Next, we examine the L1 norm of R1 . From (3.94), we find ||R1 (Φ)|| 1 ≤|| − Le Φ1 + −β 11 L N j=1 α α qj 11 φj 12 || 1 L N α α α (Φ1 11 − qj 11 )φj 12 || 1 L j=1   N N α α + −β Φ1 11 ( φj )α12 − φj 12  . j=1 j=1 L1 + −β || 73 (3.106) The first normed term above is zero by the definition of Φ1 from (3.2). We estimate the second and the third terms as we did for Φ2 . For the third term, we have that α 12 N α  φj  − φj 12 ≤ C r, j=1 j=1 L∞  N (3.107) for p ∈ Kl , so 0  −β Φα11  1 N α 12  N j=1 α φj 12  − j=1 L1 α ≤ r−β Φ1 11 1 L (3.108) ≤C r−2 , φj  (3.109) where we used (3.45). As before, we Taylor expand Φ1 in the second term, N N α11 α11 α12 α α (Φ1 − qj )φj || 1 ≤ || ∂x (Φ1 11 )(sj )|x − pj |φj 12 || 1 (3.110) || L L j=1 j=1 N α α ≤C |∂x (Φ1 11 )(sj )| |x − pj |φj 12 1 L j=1 (3.111) N ≤C α ||∂x (Φ1 11 )||L∞ (3.112) j=1 ≤ C 2−β , 74 (3.113) with sj ∈ R for each j and again we used (3.43) and (3.44). We deduce that −β || N α α α (Φ1 11 − qj 11 )φj 12 || 1 ≤ C 2−2β = C α . L j=1 (3.114) Together (3.106), (3.109), and (3.113) yield ||R1 (Φ)|| 1 ≤ C α . L (3.115) which establishes (3.95). The asymptotic formula (3.94) for the residual follows by identifying the leading order terms 75 Chapter 4 Linearized Equation and Spectrum We decompose solutions of (1.1) as    U  ∗   = Φp + W (x, t), V (4.1) where W ∗ = W + Φ1 and the pulse positions are functions of time p = p(t). We are in a sense putting a correction term into our ansatz through the term W ∗ . We choose Φ1 = Φ1,1 , Φ1,2 T such that ˜ Lp Φ1 ≡ −˜p R(Φ), π 0 0 76 (4.2) so ˜ ˜ Φ1 ≡ −L−1 πp R(Φ). p0 0 (4.3) ˜ L is defined in (4.9) as the reduced linear operator frozen at the point p0 , and π is an ˜ orthogonal spectral projection defined in (5.13). W = (W1 , W2 )T is the remainder. Inserting the decomposition (4.1) into the system (1.1) yields Wt + ∂Φ ∂ Φ1 + ∂p ∂p ˙ p = R(Φ) + Lp Φ1 + Lp W + N (Φ1 , W ), (4.4) where the residual R(Φ) was defined in (3.93) and the linearized operator Lp is defined as  −β α Φα11 Φα12 −1 e −β α Φα11 −1 Φα12 12 1 11 1  −L11 + 2 2   Lp ≡    α α −1 α −1 α 2 ∂x − 1 + α22 Φ1 21 Φ2 22 α21 Φ1 21 Φ2 22     .   (4.5) Up to constants, the nonlinearity is    N1 (Φ1 , W )  N (Φ1 , W ) ≡  , N2 (Φ1 , W ) 77 (4.6) where α −2 ∗ α α −1 α −1 ∗ ∗ N1 (Φ1 , W ) = −β Φ1 11 Φ2 12 W1 W2 + Φ1 11 Φ2 12 W2 2 α −2 α ∗ + −β Φ1 11 Φ2 12 W1 2 α α −2 ∗ α −1 α −1 ∗ ∗ N2 (Φ1 , W ) =Φ1 21 Φ2 22 W1 W2 + Φ1 21 Φ2 22 W2 2 α −2 α ∗ +Φ1 21 Φ2 22 W1 2 , (4.7) (4.8)    ∗  W   Φ1,1 + W1  with  1  =  . ∗ Φ1,2 + W2 W2  4.1 The reduced linearization To simplify the study of the spectral problem we introduce the reduced linearization to be:  ˜  L11   ˜ Lp ≡    J21   e −β J 12   −L11 0     =     ˜ ˜ J21 L22 L22     J11 J12      + −β      0 0    ,    (4.9) where 2 ˜ L22 = ∂x − I + α22 N j=1 78 α −1 φ0 22 (x − pj ). (4.10) We define the J21 component as: N J21 = α21 α −1 α qj 21 φj 22 . (4.11) j=1 The potentials J11 and J12 are finite rank projections α J11 =α11 ξ T ⊗ (Φ2 12 Qα11 −1 χ) (4.12) α −1 J12 =α12 ξ T ⊗ (Φ2 12 Qα11 χ), (4.13) where Q is the N × N diagonal matrix Qjj = qj for each j. From Weyl’s theorem on the essential spectra of compact perturbations of operators, we know that the ˜ essential spectrum of Lp and Lp coincide with that of Le : 11 ˜ σess (Lp ) = σess (Lp ) = B = − −2 k 2 − α µ|k ∈ R . (4.14) ˜ The difference Lp − Lp is large, but it will enjoy the enhanced resolvent estimate ˜ (5.17), since the difference Lp − Lp has no mass in each window χj of the partition of unity. 79 4.2 The point spectrum ˜ Proposition 4.1. The spectrum of L can be broken into three parts: an essential ˜ part B, a part from the point spectrum of L22 , and a part controlled by the finite rank perturbations: ˜ ˜ σ(Lp ) ⊂ B ∪ σp (L22 ) ∪ λ| det(I + Nλ (p)) = 0 . (4.15) The N × N matrix Nλ is given by (4.25). Proof: The following eigenvalue problem defines the point spectrum:      Ψ1   0  ˜ (Lp − λ)   =  . 0 Ψ2 (4.16) −(Le + λ)Ψ1 = − −β (J11 Ψ1 + J12 Ψ2 ) 11 (4.17) This expands to ˜ (L22 − λ)Ψ2 = − J21 Ψ1 . (4.18) ˜ ˜ Now if λ ∈ σp L22 ∪ B, then we can invert L22 − λ in the second equation / ˜ Ψ2 = −(L22 − λ)−1 J21 Ψ1 . 80 (4.19) Substituting this into the first equation and inverting Le + λ, we arrive at the 11 following scalar problem: ˜ Ψ1 = −β (Le + λ)−1 J11 − J12 (L22 − λ)−1 J21 Ψ1 . 11 (4.20) Recalling (4.12)-(4.13) we regroup the right hand side into a single, finite rank operator, Ψ1 = −β (Le + λ)−1 JlT ⊗ Jr · Ψ1 , 11 (4.21) where the left and right components of the tensor product are Jl =ξ (4.22) α α −1 ˜ Jr =(α11 Φ2 12 Qα11 −1 − α12 Φ2 12 (L22 − λ)−1 J21 Qα11 )χ. (4.23) We project (4.21) with ⊗Jr ⊗Jr · Ψ1 = −β ⊗ Jr · (Le + λ)−1 JlT ⊗ Jr · Ψ1 , 11 (4.24) and introduce the matrix Nλ (p) = − −β ⊗ Jr · (Le + λ)−1 JlT , 11 81 (4.25) so that the eigenvalue problem reduces to (I + Nλ ) ⊗ Jr · Ψ1 = 0. (4.26) If I + Nλ is invertible, then ⊗Jr · Ψ1 = 0 which from (4.21) implies Ψ1 = 0, and from (4.21), we see that Ψ2 = 0. Conversely if (I + Nλ )v = 0, then setting ⊗Jr · Ψ1 = v in (4.21) yields Ψ1 = −β (Le + λ)−1 JlT v. 11  (4.27)   Ψ1  Also Ψ2 from (4.19) yields an eigenvector Ψ =   for the eigenvalue problem Ψ2 ˜ (4.16) for λ ∈ σp L22 ∪ B. / ˜ Hence λ ∈ C\ B ∪ σp (L22 ) ˜ is an eigenvalue of L if and only if I + Nλ is invertible Proposition 4.2. Fix l0 > 0. There exists ν > 0 such that for all p ∈ Kl , 0 ˜ {Re(λ) > −ν} ∩ σp (Lp ) = σ0 (p) ∪ σf r (p), (4.28) ˜ where σ0 (p) consists of N distinct O( r ) eigenvalues which are in σp (L22 ). The set σf r is induced by the finite rank perturbations, and corresponds, up to multiplicity, 82 to the zeros of the N equations R(λ; p) = 1 α12 α21 1 α α11 φ0 12 − µj , (4.29) for all j from 1 to N , where R is an explicitly known meromorphic function on C\(−∞, −1] given by (4.43). The µj ’s are the N eigenvalues of the square matrix −β Qθ−1 GN (p). Moreover the eigenspace associated to σ0 is contained, up to λ O( r ), within the space          0   0  V = span   .  , ...,     φ φN  1 (4.30) Proof: We define the following reduced self-adjoint operator α −1 2 ˜ Lk,red = ∂x − 1 + α22 φ0 22 (x − pk ). (4.31) ˜ For the pulse separation l0 sufficiently large, we can interpret L22 as N spatially ˜ disjoint (windowed) operators. We analyze the point spectrum of Lk,red . We observe ˜ that φ0 (x−pk ) is an eigenfunction of Lk,red corresponding to the eigenvalue λ1 = 0. This follows from the fact that φk solves (2.34). We can also apply the SturmLiouville Theory to this operator. This operator has real point spectrum, and we can order the eigenvalues. Since φ0 (x − pk ) only has one zero, there is only one 83 positive eigenvalue, λ0 , the groundstate whose eigenfunction ψ0 has no zeros. This eigenvalue will also be O(1), since (4.31) does not contain an . Similarly, all negative eigenvalues will be an O(1) distance from λ1 = 0, so there exists some ν > 0 such ˜ that there are only two eigenvalues of Lk,red where λj > −ν, specifically for j = 0 or j = 1. The spectrum of each reduced operator satisfies ˜ σ Lk,red ⊂ {λ1 = 0, λ0 } ∪ (−∞, −ν]. (4.32) Now we seek to determine our representation for σf r . We simplify the inversion of ˜ (L22 − λ) on J21 . We write J21 as N well-separated pulses, each localized about the pulse positions, so the inversion simplifies to ˜ (L22 − λ)−1 J21 =α21 N α α22 α21 −1− α21 −1 α 22 ˜ (Lk,red − λ)−1 φ0 22 (x − pk ) q k k=1 +O( r ). (4.33) We introduce α ˜ Ξ0 (x − pk ) ≡(Lk,red − λ)−1 φ0 22 (x − pk ) α −1 α =(d2 − 1 + α22 φ0 22 (x − pk ) − λ)−1 φ0 22 (x − pk ), x 84 (4.34) (4.35) so that ˜ (L22 − λ)−1 J21 = α21 N α −1−α21 α22 /(α22 −1) q 21 Ξ0 (x − pk ) + O( r ). (4.36) k k=1 Neglecting the near-neighbor interactions between each localized term, and using our definition of θ from (2.67) , we may write Jr , defined in (4.23), as α α −1 ˜ Jr =(α11 Φ2 12 Qα11 −1 − α12 Φ2 12 (L22 − λ)−1 J21 Qα11 )χ N = (4.37) α α −1 q θ−1 χk α11 φ0 12 (x − pk ) − α12 α21 φ0 12 (x − pk )Ξ0 (x − pk ) . k k=1 (4.38) The functions Ξ0 (x − pk ) decay at an O(1) rate depending on the distance of λ to ˜ σess (L22 ). Since p ∈ Kl , there exists a minimal pulse separation l0 so that the 0 α12 −1 products φ0 (x−pj )Ξ0 (x−pk ) and χj Ξ0 (x−pk ) are uniformly O( r ) for r ≥ 2 when j = k. From (4.25) the (i, j) entry of the matrix Nλ is Ni,j = − −β Jri , (Le + λ)−1 Jlj 2 11 L θ−1 α φα12 (x − p ), (Le + λ)−1 ξ = − −β qi 11 0 i j L2 11 θ−1 α α φα12 −1 (x − p )Ξ (x − p ), (Le + λ)−1 ξ + −β qi 12 21 0 j i 0 i 11 85 (4.39) L2 (4.40) Applying (3.80), we have θ−1 ⊗ χ · α α φα12 −1 (x − p )Ξ (x − p ) GN ⊗ χ · ξ Ni,j = −β qi 12 21 0 i 0 i j ij θ−1 ⊗ χ · α φα12 (x − p ) GN ⊗ χ · ξ + O( 2−β ) − −β qi 11 0 j i ij θ−1 α φα12 − α α R(λ) GN + O( 2−β ), = − −β qi 11 0 12 21 ij (4.41) (4.42) where we define R(λ) ≡ α −1 Ξ0 , φ0 12 L2 . (4.43) We may represent Nλ as α Nλ = − −β α11 φ0 12 − α12 α21 R(λ) Qθ−1 GN + O( 2−β ). λ (4.44) The condition that I + Nλ has a kernel is exactly (4.29). ˜ Next we address the point spectrum of L22 . We treat this as a regular eigenvalue ˜ perturbation problem. The point spectrum of L22 consists of clusters of N eigenvalues an O( r ) distance from λ1 = 0 and λ0 , and also negative point spectrum left of −ν. We label the N eigenvalues near λ0 as λ0,k for k ∈ {1, . . . , N }. Claim 4.1. For every k, λ0,k is not an eigenvalue for our full eigenvalue problem (4.17) and (4.18). 86 Proof of claim: We have, up to O( 2−β ), that N ˜ aj φ0 (x − pj ) ∈ ker(L22 ) j=1 N ˜ bj,k ψ0 (x − pj ) ∈ ker(L22 − λ0,k ). j=1 (4.45) (4.46) We can apply Sturm-Liouville to order our eigenvalues. We arbitrarily let λ0,1 be the ground state, so the corresponding eigenfunction has no zeros. Without loss of generality, we have that bj,1 > 0 for all j. For λ0,2 , without loss of generality bj,2 > 0 for every j except for bN,2 which is negative, since the eigenfunction has exactly one zero. Due to the linear independence of each bi = (b1,i , . . . , bN,i )T for i = 1, . . . , N , this argument follows so that if we arrange each bi as a column of the matrix B, the resulting matrix is nonsingular, B = b1 , . . . , bN ∈ RN . (4.47) Consider the possibility that λ = λ0,k for every k = 1, . . . , N is an eigenvalue for ˜ Figure 4.1: This illustrates the point spectrum of L22 that is either positive or near r ) of both λ = 0 and λ . zero. There are N eigenvalues within O( 1 0 87 the eigenvalue problem (4.16). From (4.18) we have ˜ (L22 − λ0,k )Ψ2 = −J21 Ψ1 . (4.48) If we solve for Ψ1 in (4.17) where J11 and J12 are finite rank, we find Ψ1 is in the span of (Le + λ0,k )−1 ξi for i = 1, . . . , N . Thus, it is slowly varying in space. By 11 the Fredholm Alternative, −J21 Ψ1 must be orthogonal to everything contained in ˜ ker(L22 − λ0,k ). Then N O( r ) =< bj,k ψ0 (x − pj ), −J21 Ψ1 >, (4.49) Bv = O( r ), (4.50) vj =< ψ0 (x − pj ), −J21 Ψ1 > . (4.51) j=1 for each k. This is equivalent to where for each j, 88 B is nonsingular, so for each j, O( r ) = vj = < ψ0 (x − pj ), −J21 Ψ1 > N = < ψ0 (x − pj ), −α21 α −1 α Ψ1 q 21 φ 22 > k k (4.52) (4.53) k=1 α −1 α (4.54) = < ψ0 (x − pj ), −α21 Ψ1 qj 21 φj 22 > α α22 α21 −1− α21 −1 α 22 < ψ0 (x − pj ), φ0 22 (x − pj ) > ≈ − α21 Ψ1 (pj )qj (4.55) =O(1), (4.56) since Ψ1 is slowly varying, qj and Ψ1 (pj ) cannot be zero, and ψ0 (x − pj ) and α φ0 22 (x − pj ) have no zeros. Then we have a contradiction, so λ0,k cannot be an eigenvalue for (4.16) for any k With the claim proven, we continue proving the proposition. On the other hand,    0  when N = 1,   and λ1 = 0 are an eigenfunction-eigenvalue pair for our φ0 system, since Ψ2 = φ0 satisfies (4.18) for Ψ1 = 0. Also (4.17) is satisfied since 89 J12 φ0 = 0. We show this below: α −1 J12 φ0 =α12 ξ < (φ0 12 q α11 χ), φ0 > (4.57) α −1 =α12 ξq α11 −α21 (α12 −1)/(α22 −1) < φ0 12 , φ0 (x − pk ) > (4.58) =0. (4.59) For N > 1, the eigenvalue λ1 = 0 breaks into N eigenvalues of size O( r ) for sufficiently large pulse separations. At leading order, the eigenspace associated to ˜ σ0 is V = ker(L22 ). This completes the proof of Proposition 4.2 4.3 Finite rank spectrum We have characterized the finite rank spectrum σf r in terms of the matrix Nλ (p) defined in (4.25). The set σf r is the spectrum that moves as the pulse positions evolve. In order to control the evolution of this finite rank spectrum, we need p ∈ Kν to assure that σf r is bounded in the left-half complex plane away from the origin. From Proposition 4.1, σf r occurs only for λ such that I + Nλ is singular. For all p ∈ Kν , we have σf r ⊂ C ∗ , where C ∗ is appropriately contained in C ∗ ⊂ C, 90 (4.60) so that the matrix I + Nλ is invertible in a neighborhood of the contour C. Moreover as |λ| → ∞ along C, (I + Nλ ) −1 → 0. So by the continuity of (I + Nλ ), for all p ∈ Kν , there exists C > 0 such that we have the uniform bound, |(I + Nλ )−1 | ≤ C, (4.61) for all λ ∈ C and all λ to the left of C. 4.4 Adjoint eigenfunction estimates In this section we develop asymptotic expansions of the eigenfunctions Ψk N k=1 ˜ of L that correspond to the algebraically small eigenvalues, and also the adjoint † N ˜ that correspond to L† . eigenfunctions Ψ k k=1 Lemma 4.1. For p ∈ K, where K is defined in (1.13), the eigenspace corresponding to the algebraically small eigenvalues σ0 is spanned by    0  r Ψk =   + O( ), φk (4.62) for k ∈ {1, . . . , N }. The space of adjoint eigenfunctions is spanned by the set N † † (Ψ , Ψ )T , given by (4.80) and (4.70) which satisfy the following esti1,k 2,k k=1 91 mate: † † ||Ψ || 1,1 + β ||Ψ − φk || 1 ≤ C 2 , 1,k W 2,k H ξ for some C > 0 independent of (4.63) and p ∈ K. Proof: The previous proposition implies (4.62). The adjoint operator is given by:  e  −L11 J21  ˜† =  L    ˜ 0 L22    †   J11 0       −β  +  ,       † J12 0 (4.64) where α † J11 = α11 χ T Φ2 12 Qα11 −1 ⊗ ξ, (4.65) α −1 † J12 = α12 χ T Φ2 12 Qα11 ⊗ ξ. (4.66) and † † T The eigenvalue problem for Ψ† = Ψ1 , Ψ2 is   †  Ψ  ˜ (L† − λ)  1  = 0. † Ψ2 (4.67) Since we consider the small eigenvalues near zero we may neglect λ = O( r ). We 92 have the following two equations: † † † † −Le Ψ1 = − J21 Ψ2 − −β J11 Ψ1 11 † † † ˜ L22 Ψ2 = − −β J12 Ψ1 . ˜ Since φk ∈ ker L22 , we form a basis (4.68) (4.69) N † † ˜ (Ψ , Ψ )T of solutions to L† Ψ† 1,k 2,k k=1 where the second component is † † † ˜ Ψ = φk − −β L−1 J12 Ψ . 22 1,k 2,k (4.70) ˜ Using the form of L22 and that φk solves (3.3), we reduce to the following x − pk 1 ˜ φk + φ + O( r ), L−1 φk = 22 2 α22 − 1 k (4.71) ˜ which follows since L22 φk = 0 and p ∈ K. This is used to demonstrate the uniform boundedness of (4.100). Substituting (4.70) into (4.68), we have † † † † † ˜ =L−e [J21 (φk − −β L−1 J12 Ψ ) + −β J11 Ψ ] Ψ 11 22 1,k 1,k 1,k †T † † =L−e [J21 φk + −β Jr ⊗ J · Ψ ], 11 l 1,k 93 (4.72) (4.73) † where J = Jl = ξ and from (4.23), l † Jr = α α −1 ˜ α11 Φ2 12 Qα11 −1 − α12 J21 L−1 Φ2 12 Qα11 χ. 22 (4.74) † Now we project (4.73) with ⊗J , so l †T † † † † † ⊗J · Ψ = ⊗J · L−e [J21 φk + −β Jr ⊗ J · Ψ ]. 11 l 1,k l 1,k l (4.75) If we rearrange terms, we have † †T † † † (I − −β ⊗ J · L−e Jr ) ⊗ J · Ψ = ⊗J · L−e J21 φk . 11 11 l l 1,k l (4.76) † † † † ⊗J · Ψ = (I + N )−1 ⊗ J · L−e J21 φk , 11 l 1,k λ l (4.77) † † †T N = − −β ⊗ J · L−e Jr . 11 λ l (4.78) So where Next, we plug (4.77) into (4.73) and factor to obtain † † † †T Ψ =L−e [J21 φk + −β Jr (I + N )−1 ⊗ J · L−e J21 φk ] 11 11 λ l 1,k (4.79) † † †T =[I + −β L−e Jr (I + N )−1 ⊗ J ·]L−e J21 φk . 11 11 l λ (4.80) 94 Using the estimate (3.14), we have ||L−e J21 φk || 1,1 ≤ c 11 W ξ β | ⊗ χ · J φ | + 2 ||J φ || 21 k 21 k L1 1,p , (4.81) where α −1 α J21 φk = α21 q 21 φ 22 φk + O( r ). k k (4.82) Due to even-odd parity, this has algebraically small mass so | ⊗ χ · J21 φk | = O( r ). (4.83) In addition, ||J21 φk || 1 = O(1), since J21 φk is exponentially decaying away from L 1,p the pulse positions. Then −e ||L11 J21 φk || 1,1 ≤ C 2 . W ξ (4.84) Also for (4.80), we use (3.6) and have †T †T || −β L−e Jr || 1,1 ≤ C||Jr || 1 11 L W ξ ≤ C, 95 (4.85) (4.86) † 1,1 since Jr is uniformly bounded in L1 . Taking the W norm of (4.80), we have ξ †T † † † ||Ψ || 1,1 =||[I + −β L−e Jr (I + N )−1 ⊗ J ·]L−e J21 φk || 1,1 11 11 λ l 1,k W W ξ ξ (4.87) †T † † ≤C 2 + || −β L−e Jr || 1,1 ||(I + N )−1 ⊗ J · L−e J21 φk ||L∞ 11 11 λ l W ξ (4.88) † ≤C 2 + ||(I + N )−1 ||L∞ || λ ξL−e J21 φk dx 11 † ≤C 2 (1 + ||(I + N )−1 ||L∞ ), λ (4.89) (4.90) since ξj L−e J21 φk dx ≤ ||L−e J21 φk || 1,1 ≤ C 2 . 11 11 W ξ (4.91) From (4.61), we similarly have that † ||(I + N )−1 ||L∞ ≤ C, λ (4.92) † ||Ψ || 1,1 ≤ C 2 . 1,k W ξ (4.93) and conclude 96 To achieve the second part of (4.63), we use (4.70) and have † † † ˜ ||Ψ − φk || 1 =|| −β L−1 J12 Ψ || 1 22 1,k H 2,k H N α −1 α † −β α L−1 ˜ =|| χi Φ2 12 qi 11 < Ψ , ξi > || 1 12 22 1,k H i N α −1 α † ˜ < Ψ , ξi > L−1 χi Φ2 12 qi 11 || 1 =|| −β α12 22 1,k H i N α −1 α † −β ˜ < Ψ , ξi > ||L−1 χi Φ2 12 qi 11 || 1 ≤C 22 1,k H i N α −1 † ˜ ≤C −β ||Ψ || 1,1 ||L−1 χi Φ2 12 || 1 22 1,k W H ξ i ≤C 2−β , (4.94) (4.95) (4.96) (4.97) (4.98) (4.99) where as a consequence of (4.71) we have that α −1 ˜ −1 L22 χi Φ2 12 = x − pk 1 α −1 φk + φk χi Φ2 12 + O( r ), 2 α22 − 1 which is uniformly bounded in H 1 97 (4.100) Chapter 5 Resolvent and Semigroup Estimates In this chapter we generate resolvent and semigroup estimates for our reduced oper˜ ator Lp . We fix a contour C ∈ C. We define C as C = Cv ∪ C − ∪ C + , l l where Cv = C+ = l αµ − 2 + is s ∈ [−b, b] , C − = l (5.1) − i5π −ib + se 6 s ∈ [−∞, 0] , and i5π ib + se 6 s ∈ [−∞, 0] , for b positive, and independent of . We pick b ˜ ˜ sufficiently large so that L22 − λ , L22 − λ , and I + Nλ are all invertible on C. The contour C is illustrated in Figure 5.1. 98 Figure 5.1: This is an illustration of our contour C. Given F = (f1 , f2 )T and λ ∈ C, we examine the resolvent problem     g1   ˜ (Lp − λI)  = g2  f1  . f2 (5.2) We invert the equation for g2 and have ˜ g2 = (L22 − λ)−1 (f2 − J21 g1 ) . 99 (5.3) The equation for g1 is Le + λ − −β J11 g1 − −β J12 g2 = −f1 . 11 (5.4) If we substitute (5.3) into this and rearrange terms we obtain ˜ (Le + λ)g1 − −β J11 − J12 (L22 − λ)−1 J21 g1 = −KF, 11 (5.5) ˜ KF = f1 − −β J12 (L22 − λ)−1 f2 . (5.6) where Recalling Jl and Jr from (4.22) and (4.23), we simplify (5.5) to (Le + λ)g1 − −β JlT ⊗ Jr · g1 = −KF. 11 (5.7) If we invert the constant coefficient operator on the left and project with ⊗Jr , we have ⊗Jr · g1 − −β ⊗ Jr · (Le + λ)−1 JlT ⊗ Jr · g1 = − ⊗ Jr · (Le + λ)−1 KF. (5.8) 11 11 100 Recalling the matrix Nλ from (4.25), we write the above as I + Nλ ⊗ Jr · (Le + λ)−1 JlT ⊗ Jr · g1 = − ⊗ Jr · (Le + λ)−1 KF, 11 11 (5.9) and inverting we obtain an expression for the projection of g1 ⊗Jr · g1 = −(I + Nλ )−1 ⊗ Jr · (Le + λ)−1 KF. 11 (5.10) If we substitute this into (5.7) and isolate g1 , we establish the closed form expression g1 = (Le + λ)−1 11 5.1 −β J T (I + N )−1 ⊗ J · (Le + λ)−1 − I KF. r λ 11 l (5.11) Spectral projections The spectral projection associated to the N -point spectrum σ0 near zero is defined by N πp U ≡ † (U , Ψj ) Ψ , † j j=1 (Ψj , Ψj ) (5.12) and the complementary projection is defined as π p U ≡ I − πp U . ˜ 101 (5.13) Recalling || · ||X defined in (1.40), we define the associated spectral subset that is ˜ associated to the temporally decaying solutions of the semigroup generated by Lp : Xp ≡ {U ||U ||X < ∞, πp U = 0}. 5.2 (5.14) Resolvent estimates Proposition 5.1. For all λ on C, F ∈ Xp , and all p ∈ K, we have the following ˜ resolvent estimates for L, ˜ ||(L − λ)−1 F ||X ≤C 2−β 2−β Re(kλ ) 1+ |kλ | |(I + Nλ )−1 | β ||f || + ||f || 1 L1 2 L2 , (5.15) ˜ ||(L − λ)−1 F ||X ≤C 2 2−β |kλ |Re(kλ ) 2−β +C Re(kλ ) 1+ Re(kλ ) 2−β 1+ Re(kλ ) 102 |(I + Nλ )−1 | ||f1 || 1,1 W ξ |(I + Nλ )−1 | ||f2 ||L . 2 (5.16) If in addition the coarse-grained projection of f1 is small, then we have the enhanced residual estimate ˜ ||(L − λ)−1 F ||X ≤C +C 2−β Re(kλ ) 2−β Re(kλ ) (I + Nλ )−1 2 | ⊗ χ · f1 | + C 2 ||f1 || 1 L |kλ | 1,p β 2 ||f || 1 L1 + | ⊗ χ · f1 | + ||f2 ||L2 1,p . (5.17) Proof: We have that    g1  ˜ (L − λ)−1 F =  . g2 (5.18) 1,1 We apply the W norm to g1 as represented in (5.11) and also use the estimate ξ (3.6): 2−β ||g1 || 1,1 ≤C ||J T (I + Nλ )−1 ⊗ Jr · (Le + λ)−1 KF || 1 11 L Re(kλ ) l W ξ 2 +C ||KF || 1 . L Re(kλ ) (5.19) From the definition of Jl = ξ, and the fact that the L1 norm of the components of ξ are each one, we have ||JlT (I + Nλ )−1 || 1 ≤ C|(I + Nλ )−1 |. L 103 (5.20) ˜ Contained within ⊗Jr is (L22 − λ)−1 which is uniformly invertible from L2 to H 1 for λ ∈ C since F ∈ Xp . So we have: |⊗Jr ·(Le +λ)−1 KF | ≤ ||Jr ||L ||(Le +λ)−1 KF ||L∞ ≤ C 11 11 1 2 |kλ | ||KF || 1 , (5.21) L where we used (3.8) and the fact that: α α −1 ˜ ||Jr || 1 =||(α11 Φ2 12 Qα11 −1 − α12 Φ2 12 (L22 − λ)−1 J21 Qα11 )χ|| 1 (5.22) L L α α −1 ˜ ≤C||Φ2 12 χ|| 1 + C||Φ2 12 (L22 − λ)−1 J21 χ|| 1 L L α −1 ˜ ≤C 1 + ||Φ2 12 || 2 ||(L22 − λ)−1 J21 χ|| 2 L L (5.23) (5.24) ≤C 1 + ||J21 χ|| 2 L (5.25) ≤C. (5.26) Applying (5.20) and (5.21) to (5.19), we have 2 ||g1 || 1,1 ≤ C Re(kλ ) W ξ 2−β |kλ | |(I + Nλ )−1 | + 1 ||KF || 1 . L (5.27) Estimating the right hand side, we have that ˜ ||KF || 1 ≤ ||f1 || 1 + −β ||J12 (L22 − λ)−1 f2 || 1 . L L L 104 (5.28) Furthermore, ˜ ˜ ||J12 (L22 − λ)−1 f2 || 1 ≤||J12 || 2 ||(L22 − λ)−1 f2 || 2 L L L ≤C||f2 || 2 . L (5.29) (5.30) Then we have: 2−β ||g1 || 1,1 ≤ C Re(kλ ) W ξ 2−β |kλ | |(I + Nλ )−1 | + 1 β ||f || + ||f || 1 L1 2 L2 . (5.31) Next we take the H 1 norm of g2 from (5.3): ˜ ||g2 || 1 =||(L22 − λ)−1 (f2 − J21 g1 )|| 1 H H (5.32) ≤C(||f2 || 2 + ||J21 g1 || 2 ) L L (5.33) ≤C(||f2 || 2 + ||g1 || 1,1 ). L W ξ (5.34) Applying (5.31) to (5.32), and also combining these bounds, we have (5.15). For (5.16), we again apply the W 1,1 norm to g1 from (5.11) and then split the ξ 105 estimate into two terms: ||g1 || 1,1 ≤||(Le + λ)−1 −β JlT (I + Nλ )−1 ⊗ Jr · (Le + λ)−1 || 1,1 11 11 W W ξ ξ +||(Le + λ)−1 KF || 1,1 11 W ξ 2−β ≤C ||J T (I + Nλ )−1 ⊗ Jr · (Le + λ)−1 KF || 1 11 L Re(kλ ) l (5.35) +||(Le + λ)−1 KF || 1,1 , 11 W ξ (5.36) where we applied (3.6) to the first part. Addressing part of this term: | ⊗ Jr · (Le + λ)−1 KF | =| < Jr , (Le + λ)−1 KF > | 11 11 (5.37) ≤||Jr || 1 ||(Le + λ)−1 KF ||L∞ 11 L (5.38) ≤C||(Le + λ)−1 KF || 1,1 . 11 W ξ (5.39) Bounding JlT term as in (5.20), we have 2−β ||g1 || 1,1 ≤ C W ξ 1+ Re(kλ ) |(I + Nλ )−1 | ||(Le + λ)−1 KF || 1,1 . 11 W ξ 106 (5.40) Using (3.7) and (3.8) we obtain  2 2−β  ||(Le + λ)−1 KF || 1,1 ≤ C  ||f || + ||f ||  . (5.41) 11 |kλ |Re(kλ ) 1 W 1,1 Re(kλ ) 2 L2 W ξ ξ Combining these estimates, we have 2−β 2−β ||g1 || 1,1 ≤ C Re(kλ ) W ξ 1+ Re(kλ )  |(I + Nλ )−1 |  β |kλ |  ||f1 || 1,1 + ||f2 ||L  . 2 W ξ (5.42) We bound ||g2 || 1 as in (5.32) to obtain (5.16). H To obtain (5.17), we examine the case when the coarse-grained projection of f1 is small. The bound on the f2 component is the same as in (5.15), so without loss 1,1 of generality we consider the case F = (f1 , 0)T . Taking the W norm of g1 as ξ represented in (5.11), we have ||g1 || 1,1 =||(Le + λ)−1 11 W ξ −β J T (I + N )−1 ⊗ J · (Le + λ)−1 − I f || r 1 1,1 λ 11 l W ξ (5.43) ≤C −β ||(Le + λ)−1 JlT || 1,1 (I + Nλ )−1 | ⊗ Jr · (Le + λ)−1 f1 | 11 11 W ξ +C||(Le + λ)−1 f1 || 1,1 . 11 W ξ (5.44) 107 Using (3.15) and the uniform L1 bound on Jr we obtain | ⊗ Jr · (Le + λ)−1 f1 | ≤||Jr || 1 ||(Le + λ)−1 f1 ||L∞ 11 11 L 2 | ⊗ χ · f1 | + 2 ||f1 || 1 ≤C L |kλ | 1,p (5.45) . (5.46) From (3.6), we have the bound 2 2 ||JlT || 1 ≤ C . ||(Le + λ)−1 JlT || 1,1 ≤ C 11 L Re(kλ ) Re(kλ ) W ξ (5.47) Finally, applying (3.14) to the remaining term, ||(Le + λ)−1 f1 || 1,1 ≤ C 11 W ξ 2 |kλ | | ⊗ χ · f1 | + 2 ||f || Re(kλ ) Re(kλ ) 1 L1 1,p . (5.48) Combining these estimates, we have (5.17) We use the previous proposition to obtain the following estimate on Φ1 . Lemma 5.1. Fix the pulse separation l0 > 0 sufficiently large, then there exists a constant C > 0 such that for all p ∈ K, we have following estimate: ||Φ1 ||X ≤ C 2−β . 108 (5.49) Proof: We apply the resolvent estimate (5.15) to the definition (4.3) of Φ1 which yields ˜ ˜ ||Φ1 ||X =||L−1 πp R(Φ) ||X p0 0 ≤C β ||[˜ π π p0 R(Φ)]1 ||L1 + ||[˜p0 R(Φ)]2 ||L2 . (5.50) (5.51) Using the residual estimate (3.95), we have ||[˜p R(Φ)]1 ||L ≤||R1 (Φ)||L + ||[πp R(Φ)]1 ||L π 1 1 1 0 0 ≤C α + ||[πp R(Φ)]1 ||L . 1 0 (5.52) (5.53) However, we have the estimate N (R(Φ), Ψ† ) j Ψ1,j ||L ||[πp R(Φ)]1 ||L =|| † 1 1 0 (Ψj , Ψj ) j=1 N † ||R1 (Φ)|| 1 ||Ψ1,j ||L∞ ||Ψ1,j ||L ≤C L 1 j=1 N † +C ||R2 (Φ)|| 2 ||Ψ2,j || 2 ||Ψ1,j ||L L L 1 j=1 ≤C 2−β , (5.54) (5.55) (5.56) which results from our previous residual estimates and adjoint eigenvector estimates. 109 Using the residual estimate (3.96), we have ||[˜p R(Φ)]2 ||L ≤||R2 (Φ)||L + ||[πp R(Φ)]2 ||L π 2 2 2 0 0 ≤C 2−β , (5.57) (5.58) since as before N (R(Φ), Ψ† ) j Ψ2,j ||L ||[πp R(Φ)]2 ||L =|| † 2 2 0 j=1 (Ψj , Ψj ) N † ≤C ||R1 (Φ)|| 1 ||Ψ1,j ||L∞ ||Ψ2,j ||L L 2 j=1 N † ||R2 (Φ)|| 2 ||Ψ2,j || 2 ||Ψ2,j ||L +C L L 2 j=1 ≤C 2−β . (5.59) (5.60) (5.61) Since α + β = 2 − β from the assumption 1 − α/2 − β = 0, we have our result ||Φ1 ||X ≤ C β ||[˜ π 2−β π p0 R(Φ)]1 ||L1 + ||[˜p0 R(Φ)]2 ||L2 ≤ C 110 (5.62) 5.3 Semigroup estimates ˜ For fixed p ∈ K, we see from classical results, e.g. [24], since L is sectorial, we can generate its semigroup from the Laplace transform of its resolvent. With our contour ˜ C, the semigroup S associated to L is given by the contour integral S(t)F = 1 ˜ eλt (λ − L)−1 F dλ, 2πi C (5.63) where we assume that F ∈ Xp . We have the following estimates on the semigroup. Proposition 5.2. For any t0 > 0 there exists C > 0 such that for all p ∈ K, F ∈ Xp , and t ≥ t0 the semigroup satisfies α − 2 µt β ||S(t)F ||X ≤Ce ||f1 ||L + ||f2 ||L (5.64) 1 2 α α − 2 µt − 2 µt −α )||f || ln( −α )||F ||X . ||S(t)F ||X ≤Ce ln( 1 1,1 + ||f2 ||L2 ≤ Ce W ξ (5.65) If in addition the coarse-grained projection of f1 is small, then we have the improved estimate α − 2 µt ||S(t)F ||X ≤ Ce β | ⊗ χ · f | + 2 ||f || 1 1 L1 + ||f2 ||L2 1,p 111 . (5.66) Proof: Since p ∈ K and from (4.61), we have that (I + Nλ )−1 is uniformly bounded for all λ ∈ C. For each of the above semigroup estimates, we apply the appropriate resolvent estimate from Proposition 5.1, which leads to a class of integrals to bound. For (5.64) we find, ||S(t)F ||X =|| 1 ˜ eλt (λ − L)−1 F dλ||X 2πi C (5.67) ˜ ||(λ − L)−1 F ||X |eλt |dλ (5.68) ≤C ≤C C β ||f || + ||f || 1 L1 2 L2 C |eλt | 2−β 2−β Re(kλ ) 1+ |kλ | dλ. (5.69) Similarly for (5.65) we see that ||S(t)F ||X ≤C C ˜ ||(λ − L)−1 F ||X |eλt |dλ 2−β 2−β β 1+ dλ ≤C||f1 || 1,1 |eλt | Re(kλ ) Re(kλ ) |kλ | W C ξ 2−β 2−β 1+ dλ. +C||f2 ||L |eλt | 2 C Re(kλ ) Re(kλ ) 112 (5.70) (5.71) While for (5.66) we have, ||S(t)F ||X ≤C C ˜ ||(λ − L)−1 F ||X |eλt |dλ 2−β |eλt | 1 + ≤C 2 ||f1 || 1 dλ L Re(kλ ) 1,p C 2−β 2−β |eλt | 1+ +C β | ⊗ χ · f1 | Re(kλ ) |kλ | C 2−β +C||f2 ||L |eλt | dλ. 2 C Re(kλ ) (5.72) dλ (5.73) The following claim estimates these integrals: Claim 5.1. Fix the contour C as in (5.1), then for all p ∈ K, there exists C > 0 such that 2−β |eλt | C Re(kλ ) 4−2β |eλt | C |kλ |Re(kλ ) 4−2β |eλt | C Re(kλ )2 2 |eλt | C |kλ |Re(kλ ) 4−β |eλt | C |kλ |Re(kλ )2 α 1−β e− 2 µt dλ ≤C (5.74) dλ ≤C α α ln( −α ) e− 2 µt (5.75) dλ ≤C α α ln( −α ) e− 2 µt (5.76) dλ ≤C ln α −α e− 2 µt (5.77) dλ ≤C ln α −α e− 2 µt . (5.78) Proof of claim: The two angled parts of the contour C − and C + are straightl l 113 forward to estimate, because the exponential decay is dominant as Re(kλ ) → −∞. αµ Our concern is along the vertical part of the contour Cv = − 2 + is|s ∈ [−b, b] . On Cv , we have kλ = i tan−1 (s −α /µ) α µ + is = 4 2α µ2 + s2 e 2 . (5.79) Using trigonometric properties, |kλ | = Re(kλ ) = 4 2α 2 µ + s2 4 2α 2 µ + s2 cos =± 4 2α 2 µ + s2 (5.80) tan−1 (s −α /µ) 2 αµ 1/2 + 2 (5.81) . (5.82) s2 + 2α µ2 For µ = 0 we have |kλ | = Re(kλ ) = 114 |s| (5.83) |s| . 2 (5.84) Applying this to (5.74) we have, 2−β |eλt | dλ Cv Re(kλ ) α 1−β e− 2 µt − ≤ (5.85) 0 b 2 ds + |s| 0 −b 2 ds |s| (5.86) α − 2 µt √ 1−β ≤Ce b . (5.87) For (5.75), we argue that 4−2β |eλt | Cv |kλ |Re(kλ ) dλ = 2+α |eλt | dλ Cv |kλ |Re(kλ ) α − 2 µt b ≤ Ce 0 2α µ2 + s2 (5.88) α 1/2 + ds αµ 2 s2 + 2α µ2 (5.89) α − 2 µt ≤ Ce α b ds 0 α − 2 µt = Ce b ds 0 α − 2 µt α = Ce (5.90) 2α µ2 + s2 µ2 + b −α 0 115 (5.91) s 2 α d˜ s µ2 + s 2 ˜ , (5.92) s where we substituted s = α . Then for the integral in the previous line, we have ˜ b −α 0 d˜ s = sinh−1 µ2 + s 2 ˜ = sinh−1 ≤ 2 ln for b −α s ˜ µ (5.93) 0 b −α µ b −α µ , (5.94) (5.95) sufficiently small α < .475b . µ (5.96) This gives us 4−2β |eλt | Cv |kλ |Re(kλ ) α − 2 µt dλ ≤ Ce α ln( −α ) . (5.97) α ln( −α ) . (5.98) For (5.76), the bound follows as in (5.75), where 4−2β |eλt | C Re(kλ )2 α − 2 µt dλ ≤ Ce We also have that (5.77) follows like (5.75), so 2 |eλt | C |kλ |Re(kλ ) α − 2 µt dλ ≤ Ce ln( −α ). (5.99) The final integral to bound is (5.78). Again following the previous methods we have 116 that 4−β |eλt | C |kλ |Re(kλ )2 α − 2 µt dλ ≤ Ce ln( −α ) (5.100) With the claim prove, we apply the estimates on these five integrals, which leads to all three semigroup estimates. Both (5.77) and (5.78) account for the ln( −α ) in (5.65) 117 Chapter 6 Nonlinear Adiabatic Stability by Renormalization Group In this chapter we prove the adiabatic stability results in (1.23) and derive the limiting pulse dynamics in (1.24). We assume at a time t0 that our initial data U0 = (U0 , V0 )T satisfies ||U0 − Φp ||X ≤ δ, ∗ (6.1) for some δ > 0 and p∗ ∈ K. The following proposition allows us to choose a base point p0 about which we will develop a local coordinate system. Proposition 6.1. Fix δ sufficiently small. Given p∗ ∈ K for K defined in (1.13) 118 and U0 satisfying the estimate ||W∗ ||X ≤ δ for W∗ ≡ U0 − Φp , there exists M > 0 ∗ and a smooth function H : X → K such that p = p∗ + H(W∗ ) satisfies W0 ≡ U0 − Φp ∈ Xp , (6.2) for Xp defined in (5.14). Moreover, if W∗ ∈ Xp for some p ∈ K, then ˜ ˜ |p − p∗ | ≤ M1 ||W∗ ||X |p∗ − p|. ˜ (6.3) Proof: We may write U0 = Φp + W∗ and U0 = Φp + W0 , which implies that ∗ W0 = W∗ + Φp − Φp . ∗ (6.4) The equation (6.2) requires that 0 = πp W0 = πp (W∗ + Φp − Φp ), ∗ (6.5) which is equivalent to solving the system Λ(p, W∗ ) = (Λ1 , . . . , ΛN )T = 0 where † Λj (p, W∗ ) =< W∗ + Φp − Φp , Ψj >= 0. ∗ 119 (6.6) It is clear that Λ(p∗ , 0) = 0. (6.7) We examine the following gradient   † †  < ∂p1 Φp , Ψ1 > · · · < ∂pN Φp , Ψ1 >     . . .. . .  . . . p Λ|(p=p∗ ,W∗ =0) = −      † † < ∂p1 Φp , ΨN > · · · < ∂pN Φp , ΨN > . (6.8) p=p∗ Using the asymptotic reductions (3.48) and (3.79),    0  2−β ), ∂pj Φp =   + O( φj (6.9) in L∞ . Using (3.47), we have ||∂pj Φ1,p || 1 = O(1). L (6.10) T † Ψj (., pj ) = O( 2 ), φj + O( 2−β ) , (6.11) Also from (3.79), 120 in L∞ . Using these estimates, we reduce (6.8) to 2 −2α21 /(α22 −1) + O( 2−β ), p Λ|p=p∗ ,W∗ =0 = −||φ ||L2 Q (6.12) where Q is the same diagonal matrix as before. From (2.29) in Theorem 2.1, we have that q is uniformly bounded from zero. Applying the implicit function theorem, we are guaranteed the existence of a smooth function H : X → K which provides the solution W0 in a neighborhood about (p∗ , 0). Also if W∗ ∈ Xp for some p ∈ K, then by definition ˜ ˜ † p < W∗ , Ψj (˜) >= 0, (6.13) for all j from 1 to N . By the Mean Value Theorem, there exists p such that |p − p∗ || p Λ(p )| = |Λ(p, 0) − Λ(p∗ , 0)|, 121 (6.14) where we have | p Λ(p )| = O(1). Then |p − p∗ | ≤M |Λ(p∗ , 0) − Λ(p, 0)|  (6.15)  . . .       † =M 0 −  < Φ − Φ , Ψ (p∗ ) >  p∗ p j     . . .   . . .       =M  < W∗ , Ψ† (p∗ ) − Ψ† (˜) >  ,   j j p   . . . (6.16) (6.17) since Φp∗ − Φp = W0 − W∗ and πp W0 = 0. Using (6.13), the H¨lder inequality, and o the fact that the X-norm controls L∞ , we have N |p − p∗ | ≤M j=1 N ≤M j=1 † † ||Ψj (p∗ ) − Ψj (˜)|| 1 ||W∗ ||L∞ p L (6.18) † † ||Ψj (p∗ ) − Ψj (˜)|| 1 ||W∗ ||X . p L (6.19) To finish the proof, we use the Mean Value Theorem † † p |Ψj (p∗ ) − Ψj (˜)|dx = † |p∗ − p|| Ψj (p )|dx ˜ ≤C|p∗ − p|. ˜ 122 (6.20) (6.21) Redefining our constant, we have (6.3) 6.1 Projected equations We start the renormalization group procedure by freezing p = p0 in Xp , where 0 p0 is the base point provided by Proposition 6.1. Then we rewrite (4.4), inserting ˜ ∆L ≡ Lp − Lp , so that the evolution for the remainder W can be represented as 0 Wt + ∂Φ ∂ Φ1 + ∂p ∂p ˙ ˜ p = R(Φ) + Lp Φ1 + W + ∆L Φ1 + W + N (Φ1 , W ) 0 (6.22) W (x, 0) = W0 , (6.23) where W ∈ Xp and p = p(t). To examine ∆L, we break it into secular and reductive 0 parts, ∆L = ∆s L + ∆r L, where ∆s L ≡ Lp − Lp 0 (6.24) ˜ ∆r L ≡ Lp − Lp . 0 0 (6.25) 123 ˜ We recognize from the definition of Φ1 in (4.3) that πp Lp Φ1 = 0, since πp and 0 0 0 πp are orthogonal projections. Similarly, we have ˜ 0 ∂ Φ1 ˙ πp p = 0, 0 ∂p (6.26) since ∂ Φ1 † p˙j , Ψj ∂pj ∂ Φ1 p˙ = πp 0 ∂pj j † Ψj , Ψj ∂ Φ1 † , Ψj ∂pj = † Ψj , Ψj Ψj (6.27) Ψj p˙j (6.28) ∂ Φ1 =πp p˙ 0 ∂pj j ∂ = πp Φ1 p˙j 0 ∂pj =0 (6.29) (6.30) (6.31) ˜ since πp and Lp commute, p˙j is independent of x, and πp is frozen, so the 0 0 0 differential term can be removed. We impose the non-degeneracy condition W ∈ Xp , which also implies πp Wt = 0, since πp is independent of time. Since W ∈ 0 0 0 Xp , by definition we have 0 πp W = 0. 0 124 (6.32) ˜ ˜ Also we have πp Lp W = Lp πp W = 0. Projecting both sides of (6.22) by πp , 0 0 0 0 0 and applying the non-degeneracy condition, we have the following N equations: ∂Φ ˙ † † p, Ψj = R(Φ) + ∆L Φ1 + W + N (Φ1 , W ), Ψj 2 . ∂p L L2 (6.33) ∂Φ1 † is Using the L∞ estimate on Ψj,1 from (4.63), and (3.47), which assures that ∂pj O(1) in L1 , then for the left side of (6.33) we have ∂Φ ˙ † ˙ p, Ψj = −||φ0 ||2 2 Q−2α21 /(α22 −1) + O( 2 ) p. L ∂p 2 L (6.34) Then (6.33) is equivalent to  † ˜    (R(Φ) + ∆LW + N (Φ1 , W ), Ψ1 )L2 −2α21  . ˙  −||φ ||2 Q α22 −1 + O( 2 ) p =  . .  L2  † ˜ (R(Φ) + ∆LW + N (Φ1 , W ), ΨN ) 2 L     ,   (6.35) ˜ where W = W + Φ1 . Returning to (6.22), we apply the complimentary spectral projection. From our definition of Φ1 in (4.3), we have ˜ ˜ ˜ πp R + Lp Φ1 = πp R − πp R = πp R − R + πp R = 0. ˜ ˜ 0 0 0 0 0 0 125 (6.36) So the evolution is given by ˜ ˜ Wt = R + Lp W + πp ∆L W + Φ1 + N (Φ1 , W ) ˜ 0 0 (6.37) W (x, 0) = W0 , ˜ where R = −˜p π 0 6.2 (6.38) ∂ Φ1 ˙ ∂Φ ˙ p − p is the temporal component of the residual. ∂p ∂p Decay of the remainder In this section we establish uniform estimates on the decay of ||W ||X over the duration of each renormalization interval. We introduce the following two quantities: T1 (t) = α µ(s−t0 ) sup e 2 ||W (s)||X t0 0 such that ˜ ||[∆s LW ]1 || 1 ≤C −β T2 (t) ||W ||X + 2−β L (6.42) ˜ ||[∆s LW ]2 || 2 ≤CT2 (t) ||W ||X + 2−β L (6.43) | ⊗ χ · [∆r L(W + Φ1 )]1 | ≤C 2−β ||W ||X + 2−β (6.44) ||[∆Lr (W + Φ1 )]1 || 1 ≤C −β ||W ||X + 2−β L 1,p (6.45) ||[∆Lr (W + Φ1 )]2 || 2 ≤C 2−β ||W ||X + 2−β L (6.46) ||N1 (Φ1 , W )|| 2 ≤C ||W ||2 + 4−2β X L (6.47) ||N2 (Φ1 , W )|| 2 ≤C ||W ||2 + 4−2β . X L (6.48) Proof: As before, we break ∆L into secular and reductive parts. We first examine the secular term ∆s L: α −1 α α −1 α ˜ ˜ [∆s LW ]1 = −β α11 Φ 11 Φ 12 − Φ 11 Φ 12 W1 p,1 p,2 p0 ,1 p0 ,2 α α −1 α α −1 ˜ + −β α12 Φ 11 Φ 12 − Φ 11 Φ 12 W2 p,1 p,2 p0 ,1 p0 ,2 α −1 α α −1 α ˜ ˜ [∆s LW ]2 =α21 Φ 21 Φ 22 − Φ 21 Φ 22 W1 p,2 p0 ,1 p0 ,2 p,1 α α −1 α α −1 ˜ +α22 Φ 21 Φ 22 − Φ 21 Φ 22 W2 . p,1 p,2 p0 ,1 p0 ,2 127 (6.49) (6.50) The Φ2 terms are rapidly decaying away from each pulse position and Φ1 is slowly varying. In particular, the Φ2 terms are Lipschitz in p with an O(1) constant while the slowly varying Φ1 terms are Lipschitz in p with a small constant. However, Φ1 always appears multiplied by Φ2 , and the larger Lipschitz constant prevails. In particular, we estimate α α α α ||Φ 22 − Φ 22 || 1 + ||Φ 22 − Φ 22 || 1 ≤ C|p − p0 |. p0 ,2 L p,2 p0 ,2 H p,2 (6.51) Recalling the definition of T2 from (6.40), we have α −1 α α −1 α ˜ ˜ ||[∆s LW ]1 || 1 ≤C −β ||Φ 11 Φ 12 − Φ 11 Φ 12 || 1 ||W1 ||L∞ p,1 p,2 p0 ,1 p0 ,2 L L α α −1 α α −1 ˜ +C −β ||(Φ 11 Φ 12 − Φ 11 Φ 12 )|| 2 ||W2 || 2 p,1 p,2 p0 ,1 p0 ,2 L L ≤C −β T2 (t) ||W ||X + 2−β , (6.52) (6.53) where ˜ ||W1 ||L∞ ≤ ||W1 ||L∞ + ||Φ1,1 ||L∞ ≤ ||W ||X + C 2−β , (6.54) using the bound (5.49). Then we have established (6.42). The estimate (6.43) follows in the same manner, where ˜ ||[∆s LW ]2 || 2 ≤ CT2 (t) ||W ||X + 2−β . L 128 (6.55) ˜ Next we examine the reductive term. The difference Lp − Lp is large, but 0 0 we are able to estimate this difference with the enhanced residual estimate (5.17). We choose the reduced linearization so it determines average values over each pulse region, so the differences of the two operators have little mass over each pulse region. The reductive term takes the form α −1 α ˜ ˜ [∆r LW ]1 = −β α11 Φ 11 Φ 12 − J11 (p0 ) W1 p0 ,2 p0 ,1 α α −1 ˜ + −β α12 Φ 11 Φ 12 − J12 (p0 ) W2 p0 ,1 p0 ,2 α −1 α ˜ ˜ [∆r LW ]2 = α21 Φ 21 Φ 22 − J21 (p0 ) W1 p0 ,1 p0 ,2   N α α −1 α −1 ˜ +α22 Φ 21 Φ 22 − φ0 22 (x − pj,0 ) W2 . p0 ,1 p0 ,2 j=1 (6.56) (6.57) To show the estimate (6.44), we first define ˜ ˜ ˜ χj [∆r LW ]1 dx = L(W1 ) + R(W2 ), (6.58) where ˜ L(W1 ) = −β χj α −1 α ˜ α11 Φ 11 Φ 12 − J11 (p0 ) W1 dx, p0 ,1 p0 ,2 (6.59) ˜ R(W2 ) = −β χj α α −1 ˜ − J12 (p0 ) W2 dx. α12 Φ 11 Φ 12 p0 ,1 p0 ,2 (6.60) and 129 ˜ ˜ Next, we expand the potentials J11 and J12 underneath L(W1 ) and R(W2 ) respec˜ tively. For brevity, we only expand the R(W2 ) term.  ˜ R(W2 ) = −β α12  α α −1 ˜ χj Φ 11 Φ 12 W2 − p0 ,1 p0 ,2 j α −1 α ˜ χj Φ 12 qj 11 W2 dzξj  dx p0 ,2 (6.61) = −β α12 α −1 ˜ α α χj Φ 11 Φ 12 W2 dx − qj 11 p0 ,1 p0 ,2 α −1 ˜ χj Φ 12 W2 dx , p0 ,2 (6.62) since each ξj is mass one, and χj windows all ξk to zero except for k = j. Now we Taylor expand Φ1 at each pulse location and use the fact that ∂x Φ1 = O( 2−β ) in ˜ L∞ near each pulse position. Again, for brevity, we continue with only the R(W2 ) term: ˜ R(W2 ) = −β α12 α α χj qj 11 + ∂x Φ1 11 (p ) p0,j − pj α − −β α12 qj 11 α −1 ˜ Φ 12 W2 dx p0 ,2 α −1 ˜ χj Φ 12 W2 dx p0 ,2 α −1 ˜ χj ∂x Φα11 (p ) p0,j − pj Φ 12 W2 dx p0 ,2 α −1 ˜ ≤C 2−β ||χj Φ 12 || 1 ||W2 ||L∞ p0 ,2 L ≤C −β ≤C 2−β ||W ||X + 2−β . (6.63) (6.64) (6.65) (6.66) 130 We have that | ⊗ χ · [∆r L(W + Φ1 )]1 | ≤ C 2−β ||W ||X + 2−β . (6.67) Next we bound (6.45), α −1 α ˜ ˜ ||[∆Lr W ]1 || 1 ≤C −β || α11 Φ 11 Φ 12 − J11 (p0 ) W1 || 1 p0 ,1 p0 ,2 L L 1,p 1,p α α −1 ˜ +C −β || α12 Φ 11 Φ 12 − J12 (p0 ) W2 || 1 . p0 ,1 p0 ,2 L 1,p (6.68) Since both terms follow similarly, for brevity we examine the first term: α −1 α ˜ C −β || α11 Φ 11 Φ 12 − J11 (p0 ) W1 || 1 ≤ C −β p0 ,1 p0 ,2 L 1,p N ˜ Dj (W1 ), (6.69) j=1 where we notate ˜ Dj (W1 ) = α −1 α ˜ χj |(1 + |x − pj |) α11 Φ 11 Φ 12 − J11 (p0 ) W1 |dx. p0 ,1 p0 ,2 131 (6.70) Under the summation for any j: ˜ Dj (W1 ) ≤ + α −1 α ˜ χj |(1 + |x − pj |)Φ 11 Φ 12 W1 |dx p0 ,1 p0 ,2 α −1 ˜ α χj |(1 + |x − pj |) χi Φ2 12 qi 11 W1 dzξi dx α −1 α ˜ ≤C||χj (1 + |x − pj |)Φ 11 Φ 12 || 1 ||W1 ||L∞ p0 ,1 p0 ,2 L α ˜ +C||χj Φ2 12 || 1 ||W1 ||L∞ |χj (1 + |x − pj |)ξj |dx L ≤C ||W ||X + 2−β , (6.71) (6.72) (6.73) where the linear growth term (1+|x−pj |) is dominated by the exponentially decaying terms in each norm. Connecting the previous estimates we have: ||[∆Lr (W + Φ1 )]1 || 1 ≤ C −β ||W ||X + 2−β . L 1,p (6.74) Finally, we bound (6.46) ˜ ˜ ˜ ||[∆Lr W ]2 || 2 ≤ I(W1 ) + J(W2 ), L (6.75) α −1 α ˜ ˜ I(W1 ) = ||α21 Φ 21 Φ 22 − J21 (p0 ) W1 || 2 , p0 ,1 p0 ,2 L (6.76) where we notate 132 and  α −1 α ˜ − J(W2 ) = ||α22 Φ 21 Φ 22 p0 ,1 p0 ,2 N j=1  α −1 ˜ φ0 22 (x − pj,0 ) W2 || 2 . L (6.77) For the first term, we again Taylor expand Φ1 underneath the summation:   α −1  ˜ I(W1 ) =|| α21 Φ 21 p0 ,1 N  α φj 22 + O( r ) − α21 j=1 N j=1  α −1 α ˜ qj 21 φj 22  W1 || 2 L (6.78)  N ≤C||  j=1  α −1 α ˜ ∂x Φ1 21 (pj ) p0,j − pj φj 22  W1 || 2 L α −1 ≤C||∂x Φ1 21 ||L∞ || N j=1 α ˜ p0,j − pj φj 22 W1 || 2 L ˜ ≤C 2−β ||W1 ||L∞ . (6.79) (6.80) (6.81) 133 Now we bound the other term again Taylor expanding Φ1 about the pulse location:  N N  α α −1 −α α −1 ˜ ˜ J(W2 ) =|| Φ 21 φ0 22 qj 21 − φ0 22  W2 || 2 p0 ,1 L j=1 j=1   α N N qj 21 + O( 2−β ) α −1 ˜ α −1 − φ0 22  W2 || 2 =||  φ0 22 α21 L qj j=1 j=1 N α −1 2−β || ˜ ≤C φ0 22 || 2 ||W2 ||L∞ L j=1 ≤C 2−β ||W ||X + 2−β . (6.82) (6.83) (6.84) (6.85) This gives us the following estimate: ||[∆Lr (W + Φ1 )]2 || 2 ≤ C 2−β ||W ||X + 2−β . L 134 (6.86) We estimate the first estimate on the nonlinearity in (6.47), α −1 α −1 ˜ ˜ ||N1 (Φ1 , W )|| 1 ≤C −β ||Φ1 11 Φ2 12 W1 W2 || 1 L L α11 α12 −2 ˜ 2 W2 || 1 +C −β ||Φ1 Φ2 L α −2 α ˜ 2 +C −β ||Φ1 11 Φ2 12 W1 || 1 || L (6.87) ˜ ˜ ˜ ˜ ≤C −β ||W1 ||L∞ ||W2 ||L∞ + ||W2 ||2 ∞ + ||W1 ||2 ∞ L L (6.88) ˜ ≤C 2−β ||W ||2 X (6.89) ≤C 2−β ||W ||2 + ||Φ1 ||X ||W ||X + ||Φ1 ||2 X X (6.90) ≤C 2−β ||W ||2 + ||Φ1 ||2 X X (6.91) ≤C 2−β ||W ||2 + 4−2β . X (6.92) The next estimate follows in the same manner ||N2 (Φ1 , W )|| 2 ≤ C ||W ||2 + 4−2β X L (6.93) Then with these estimates, we have the following corollary. Corollary 6.1. Fix l0 sufficiently large, then for all p ∈ K, there exists C > 0 such 135 that α ˜ || ≤CT (t)e− 2 µ(t−s) ||W || + 2−β ||S(t − s) πp ∆s LW X ˜ 2 X 0 α − µ(t−s) 2−β ˜ ||S(t − s) πp ∆r LW ||X ≤Ce 2 ˜ ||W ||X + 2−β 0 α − 2 µ(t−s) ||S(t − s) πp N (Φ1 , W ) ||X ≤Ce ˜ ||W ||2 + 4−2β . X 0 (6.94) (6.95) (6.96) Proof: The estimate (6.94) is achieved by using the semigroup estimate (5.64) and the estimates (6.42) and (6.43), where α ˜ || ≤Ce− 2 µ(t−s) ||S(t − s) πp ∆s LW X ˜ 0 β ||[∆ LW ] || + ||[∆ LW ] || s ˜ 1 L1 s ˜ 2 L2 (6.97) α − 2 µ(t−s) ≤Ce ||W ||X + 2−β . (6.98) To achieve (6.95), we apply (5.66) and recall the estimates (6.44), (6.45), and (6.46). 136 This leads to α ˜ || ≤Ce− 2 µt β | ⊗ χ · [∆r L(W + Φ )] | ||S(t − s) πp ∆r LW X ˜ 1 1 0 α − µt 2 +Ce 2 ||[∆Lr (W + Φ1 )]1 || 1 L 1,p α − µt +Ce 2 ||[∆Lr (W + Φ1 )]2 ||L 2 α − µt 2−β ≤Ce 2 ||W ||X + 2−β . (6.99) (6.100) We apply (5.66) to (6.96), recalling the estimates (6.47) and (6.48), α ˜ ) || ≤Ce− 2 µ(t−s) ||S(t − s) πp N (W ˜ X 0 β ||N (W )|| + ||N (W )|| ˜ ˜ 1 2 L1 L2 (6.101) α − 2 µ(t−s) ||W ||2 + 4−2β ≤Ce X (6.102) We return to the previous projected equation in (6.35). The next lemma estimates terms on the right side of (6.35). 137 Lemma 6.2. We have the following estimates for any j: † R1 , Ψ1,j 2 ≤C 4−2β L † [∆L Φ1 + W ]1 , Ψ1,j 2 ≤C 2−2β ||W ||X + 2−β L † N1 (Φ1 , W ), Ψ1,j 2 ≤C 2−β ||W ||2 + 4−2β . X L (6.103) (6.104) (6.105) Proof: For (6.103), we use (3.8) and have † † R1 , Ψ1,j 2 ≤ ||R1 || 1 ||Ψ1,j ||L∞ ≤ C 4−2β . L L (6.106) For (6.104), we write ∆L = ∆s L + ∆r L, and estimate the secular terms first. We use the estimate (6.42), † ˜ [∆s LW ]1 , Ψ1,j † ˜ 2 ≤||[∆s LW ]1 ||L1 ||Ψ1,j ||L∞ L (6.107) ≤C 2−β ||W ||X + 2−β . (6.108) 138 To complete the estimate of (6.104), we apply the estimate (6.45), † ˜ [∆r LW ]1 , Ψ1,j † ˜ 2 ≤C||[∆r LW ]1 ||L1 ||Ψ1,j ||L∞ L (6.109) ˜ ≤C 2−β ||[∆r LW ]1 || 1 L 1,p (6.110) ≤C 2−2β ||W ||X + 2−β . (6.111) To estimate (6.105), we use the estimate (6.47), † † N1 (Φ1 , W ), Ψ1,j 2 ≤||N1 (Φ1 , W )|| 1 ||Ψ1,j ||L∞ L L ≤C 2−β ||W ||2 + 4−2β X (6.112) (6.113) † Applying the previous lemma and using (4.63) to expand Ψ2,j in (6.33), we have the following equations for the evolution of the pulse position for each j: pj = − ˙ R2 (Φ) + [∆L(W + Φ1 )]2 + N2 (Φ1 , W ), φj 2 L 1 + O( 2 ) −2α21 /(α22 −1) qj ||φ0 ||2 2 L +O( 4−2β , 2−2β ||W ||X , 2−β ||W ||2 ). X Finally, we prove the following reduced residual estimate: 139 (6.114) Lemma 6.3. There exists C > 0 such that α 2 e− 2 µ(t−s) . ˜ ||S(t − s)˜p R||X ≤ C π 0 (6.115) Proof: Using (5.64), we have α ∂ Φ1 ˙ ∂Φ ˙ β e− 2 µ(t−s) ||[˜ ˜ ||S(t − s)˜p R||X ≤C π πp p + p] || 0 0 ∂p ∂p 1 L1 α ∂ Φ1 ˙ ∂Φ ˙ − 2 µ(t−s) +Ce ||[˜p π p + p] || . 0 ∂p ∂p 2 L2 (6.116) ∂ Φ1 ˙ We assume that p is O( 2−β ) (we justify this later). First we examine the ∂p terms, where by the definition of Φ1 , ∂ Φ1 ∂ ˜ = − L−1 πp R(Φ) 0 ∂p ∂p p0 ∂R(Φ) ˜ = − L−1 πp , p0 0 ∂p (6.117) (6.118) where we can pull the differential through the linear operator and the projection because they are both frozen at p = p0 . Also, we have T N ∂Φ N ∂Φ ∂ Φ1 ˙  1,1 1,2  p= p˙ , p˙ . ∂p ∂pj j ∂pj j j=1 j=1  140 (6.119) ˙ Then using our assumption on the size of p, β ||[ ∂ Φ1 p] || + ||[ ∂ Φ1 p] || ˙ ˙ ≤ ∂p 1 L1 ∂p 2 L2 N ∂Φ ∂Φ ( β ||[ 1 ]1 || 1 + ||[ 1 ]2 || 2 )||p˙j ||L∞ L L ∂pj ∂pj j=1 (6.120) ≤C 2 , (6.121) if for each j from 1 to N : ||[ ∂ Φ1 ] || ≤C ∂pj 1 L1 (6.122) ||[ ∂ Φ1 ] || ≤C β . ∂pj 2 L2 (6.123) Examining the first of the two inequalities above, and using (3.12): ||[ ∂ Φ1 ∂R(Φ) ∂R(Φ) ˜ ]1 || 1 = || L−1 πp || 1 ≤ C −α || πp || 1 , (6.124) p0 0 ∂pj L L L 0 ∂pj ∂pj 1 1 141 where ∂R(Φ) || πp 0 ∂pj N 1 || 1 ≤C|| L ∂R1 (Φ) † , Ψi,1 ∂pj i=1 + ∂R2 (Φ) † , Ψi,2 ∂pj Ψi,1 || 1 L (6.125) N || ∂R1 (Φ) † || 1 ||Ψi,1 ||L∞ ||Ψi,1 || 1 L L ∂pj || ≤C ∂R2 (Φ) † || 2 ||Ψi,2 || 2 ||Ψi,1 || 1 L L L ∂pj i=1 N +C i=1 ∂R (Φ) ∂R (Φ) || 1 + || 2 || 2 . ≤C 2 || 1 L L ∂pj ∂pj (6.126) (6.127) By expanding the definition of the residuals underneath the norm above, we have  ∂pk R1 (Φ) =∂pk −Le Φ1 + −β 11  N  α α qj 11 φj 12  j=1  N α α α (Φ1 11 − qj 11 )φj 12  j=1    N N α α + −β ∂p Φ1 11 ( φj )α12 − φj 12  . k j=1 j=1 + −β ∂pk  (6.128) Similar to the proof of the residual estimates in Chapter 3, the first term is 0. Since everything in the third term is smooth and continuous, it follows as in the previous residual estimates that this term is O( r ) for r ≥ 2. The middle term remains, where 142 as in the residual estimates, we Taylor expand about pj (here denotes the derivative with respect to x): α α α α (Φ1 11 − qj 11 )φj 12 = (x − pj ) Φ1 11 α (pj )φj 12 + h.o.t.. (6.129) Then at leading order,  ∂p  k  N  N  α α α α (Φ1 11 − qj 11 )φj 12  =∂p  (x − pj ) Φ1 11 k j=1 j=1 α (pj )φj 12  (6.130) α (x − pk ) Φ1 11 =∂p k α (pk )φ 12 k α (pk )φ 12 k α α +(x − pk )∂p Φ1 11 (pk )φ 12 k k α α +(x − pk ) Φ1 11 (pk )∂p φ 12 . k k (6.131) α = − Φ1 11 (6.132) (6.133) By (3.44), we have α || Φ1 11 α α (pk )φ 12 || 1 ≤ || Φ1 11 k L α (pk )||L∞ ||φ 12 || 1 ≤ C 2−β . k L 143 (6.134) For the third term in (6.133), we know from (3.79) that in any Lp norm ∂pk φk = −φk + O( 2−β ). (6.135) Then α || Φ1 11 α (pk )||L∞ ||(x − pk ) Φ1 11 α (pk )∂p φ 12 || 1 ≤ C 2−β , k k L (6.136) α since the exponential decay in ∂p φ 12 dominates any linear growth. For the second k k term in (6.132), we use (3.48), so we have α ||(x − pk )∂pk Φ1 11 α α (pk )φ 12 || 1 ≤||∂pk Φ1 11 k L α (pk )||L∞ ||(x − pk )φ 12 || 1 k L (6.137) ≤C 2−β . (6.138) Combining these estimates in (6.128), we have ||∂p R1 (Φ)|| 1 ≤ C α . k L (6.139) The estimate for the R2 term follows similar to the above, where ||∂pk R2 (Φ)|| 1 ≤ C 2−β . L 144 (6.140) Combining these estimates, we have ∂R(Φ) || πp || 1 ≤ C 2−β . L 0 ∂pj 1 (6.141) Then combining these two estimates with (6.125) and (6.124), we have achieved a stronger condition then (6.122): ||[ ∂ Φ1 ]1 || 1 ≤ C β . L ∂pj (6.142) Next we establish (6.123). Using (3.13): ||[ ∂ Φ1 ∂R(Φ) ∂R(Φ) ˜ ]2 || 2 = || L−1 πp || 2 ≤ C −α || πp || 2 . (6.143) p0 0 ∂pj L L L 0 ∂pj ∂pj 2 2 Similar to the previous term ∂R(Φ) || πp 0 ∂pj N 2 || 2 ≤C L +C || ∂R1 (Φ) † || 1 ||Ψi,1 ||L∞ ||Ψi,2 || 2 L L ∂pj || ∂R2 (Φ) † || 2 ||Ψi,2 || 2 ||Ψi,2 || 2 L L L ∂pj i=1 N i=1 ∂R (Φ) ∂R (Φ) ≤C 2 || 1 || 1 + || 2 || 2 . L L ∂pj ∂pj 145 (6.144) (6.145) The using (6.139) and (6.140), we have (6.123): ||[ Now we examine the ∂ Φ1 ] || ≤ C β. ∂pj 2 L2 (6.146) ∂Φ terms: ∂p ∂Φ ˙ ∂Φ ˙ ∂Φ ˙ ||[˜p π p]1 || 1 =|| 1 p − [πp p] || (6.147) L 0 ∂p 0 ∂p 1 L1 ∂p N ∂Φ ˙ ∂Φ ˙ † || 1 p|| 1 ||Ψ1,j ||L∞ ||Ψ1,j || 1 ≤|| 1 p|| 1 + C L L L ∂p ∂p j=1 N ∂Φ ˙ † (6.148) +C || 2 p|| 2 ||Ψ2,j || 2 ||Ψ1,j || 1 L L L ∂p j=1 ∂Φ ˙ ∂Φ ˙ ≤C(1 + 2 )|| 1 p|| 1 + C|| 2 p|| 2 L L ∂p ∂p N ∂Φ ∂Φ ≤C || 1 p˙j || 1 + || 2 p˙j || 2 L L ∂pj ∂pj j=1 N ∂Φ ∂Φ || 1 || 1 + || 2 || 2 ||p˙j ||L∞ ≤C ∂pj L ∂pj L j=1 ≤C 2−β , (6.149) (6.150) (6.151) (6.152) using (3.47), (3.79), the form of the small eigenfunctions in (4.62), and the estimates (4.63) on the adjoint eigenfunctions. From (3.79) and (4.62), we have in any Lp 146 norm, ∂Φ2 = −Ψ2,j + O( 2−β ). ∂pj (6.153) Using this, we have ∂Φ ˙ ∂Φ ˙ ∂Φ ˙ [˜p π p]2 = 2 p − [πp p] 0 ∂p 0 ∂p 2 ∂p N =− N Ψ2,j pj − ˙ j=1 (6.154) ∂Φ1 ˙ † p, Ψ1,j ∂p + ∂Φ2 ˙ † p, Ψ2,j ∂p † Ψ2,j , Ψ2,j j=1 Ψ2,j (6.155) ∂Φ1 ˙ † p, Ψ1,j ∂p Ψ2,j pj + ˙ =− Ψ2,j † Ψ2,j , Ψ2,j j=1 N Ψ − O( 2−β ), Ψ† ˙ 2,j 2,j pj Ψ2,j + † Ψ2,j , Ψ2,j j=1 ∂Φ1 ˙ † † p, Ψ1,j N ˙ 1, Ψ2,j pj ∂p =− Ψ2,j + O( 2−β ) Ψ2,j . † † Ψ2,j , Ψ2,j Ψ2,j , Ψ2,j j=1 N 147 (6.156) (6.157) Then once again in the L2 norm ∂Φ1 ˙ † p, Ψ1,j ∂Φ ˙ ∂p ||[˜p π p] || ≤|| Ψ2,j || 2 † L ∂p 2 L2 Ψ2,j , Ψ2,j j=1 † 1, Ψ2,j pj ˙ 2−β || Ψ2,j || 2 + † L Ψ2,j , Ψ2,j N ∂Φ ˙ † ≤C || 1 p|| 1 ||Ψ1,j ||L∞ ||Ψ2,j || 2 L L ∂p j=1 N (6.158) † + 2−β ||Ψ2,j || 1 ||pj ||L∞ ||Ψ2,j || 2 ˙ L L (6.159) ≤C 4−β . (6.160) If we combine these results, we establish this lemma To develop estimates on ||W ||X , we take the X-norm of (6.41) and use the semigroup estimate (5.65) on the initial term. We obtain α − 2 µ(t−t0 ) ln( −α )||W (t0 )||X + C ||W (t)||X ≤Ce t − α µ(t−s) e 2 +C t0 2 + T (t) + 2−β 2 t − α µ(t−s) e 2 ||W ||2 ds X t0 2−β + ||W || X ds, (6.161) where T2 is defined in (6.40). We evaluate the previous line at t = t , multiply by α µ(t −t0 ) , and take the sup over t ∈ (t0 , t). Recalling the definition of T1 in e2 148 (6.39), we find T1 (t) ≤C | ln( −α )|T1 (t0 ) + t +C t0 ≤C +C t t0 α 2−β (T (t) + β )e 2 µ(s−t0 ) ds 2 α 2−β T (t) + T (t)2 e 2 µ(t0 −s) ds T2 (t) + 1 1 (6.162) α −α )|T (t ) + 2−β−α T (t) + β e 2 µ(∆t) | ln( 2 1 0 T2 (t) + 2−β ∆tT1 (t) + −α T1 (t)2 , (6.163) where ∆t = t − t0 . The following lemma allows us to bound T2 in terms of T1 , ∆t, and . Lemma 6.4. 2 T2 (t) ≤ C1 ( 2−β ∆t + 2 + −α T1 ). (6.164) Proof: Applying the Mean Value Theorem, to the definition (6.40) of T2 , we find T2 (t) ≤ t0 +∆t ˙ |p(s)|ds. t0 (6.165) Recalling (6.114), the residual estimate from (3.96), the estimates on [∆L(W + 149 Φ1 )]2 in (6.43) and (6.46), and the estimate on N2 (Φ1 , W ) in (6.48), we have T2 (t) ≤C +C t0 +∆t t0 t0 +∆t t0 ≤C α ||R2 (Φ)|| 2 + T1 (t)2 e µ(t0 −s) ds L α 2−β (e 2 µ(t0 −s) T (t) + 2−β )ds T2 (t) + 1 (6.166) −α T (t) + 2−β ∆t + −α T (t)2 . 1 1 2−β ∆t + T (t) + 2−β 2 (6.167) To control T2 , we impose the following constraint on T1 α , T1 (t) ≤ 2C (6.168) where the constant C, from the right side of (6.167), is independent of p ∈ K and , but may depend upon l0 , the minimal pulse separation. We will show that the set α t T1 (t) ≤ 2C , (6.169) is a forward invariant set under the flow. Using this constraint (6.168), the −α T1 (t)T2 (t) term may be subtracted from 150 both sides, and multiplying by 2, we have T2 (t) ≤ C 2−β ∆t + 2−β ∆tT (t) + β T (t) + 4−2β ∆t + −α T (t)2 . 2 1 1 (6.170) Similarly, for ∆t satisfying the constraint β−2 ∆t ≤ 2C , (6.171) we can remove the 2−β ∆tT2 (t) term and eliminate T2 from the right side of (6.170). Using Young’s inequality on the β T1 (t) term, that is β T (t) = β+α/2 −α/2 T (t) = 1 1 −α/2 T (t) ≤ C 1 2 + −α T (t)2 , 1 (6.172) we arrive at the statement of the lemma T2 (t) ≤ C1 2−β ∆t + 2 + −α T (t)2 1 (6.173) Applying Lemma 6.4 to (6.163), we obtain T1 (t) ≤C +C α −α )|T (t ) + β ( 2−β ∆t + 2 + −α T (t)2 + β )e 2 µ(∆t) | ln( 1 0 1 −α T (t)2 + 2−β (1 + ∆t) ∆tT (t) + −α T (t)2 . 1 1 1 151 (6.174) To control T1 , we fix 0 < ω < 1 and impose an additional, more exigent, constraint on ∆t ∆t ≤ 2 −α ln µ −ωβ , (6.175) so that, in particular α 2 µ(∆t) ≤ −ωβ , e (6.176) ∆tT1 = O (| ln |) . (6.177) and from (6.168), Imposing this condition, and keeping only leading order terms in (6.174) we obtain T1 (t) ≤ C| ln | T1 (t0 ) + β + −α T1 (t)2 . (6.178) The corresponding quadratic equation in r, g(r) = C| ln | T1 (t0 ) + β − r + C| ln | −α r2 , has two positive roots 0 < r1 < r2 if (6.179) is sufficiently small, if β − α > 0, 152 (6.180) and if −α T (t )| ln |2 1 0 (6.181) is sufficiently small. In light of the assumption that 1 − α/2 − β = 0 from (1.20), then (6.180) implies α < 2/3. (6.182) Figure 6.1: This illustrates the quadratic function g(r), where either the remainder starts smaller than r1 and stays small or it begins larger the r2 . The middle interval (r1 , r2 ) is forbidden. The constraint (6.181) becomes T1 (0) ≤ C0 α , | ln( )|2 153 (6.183) for C0 sufficiently small, but independent of and p ∈ K. Under these conditions, the smaller root at leading order is r1 = C| ln | T1 (t0 ) + β , (6.184) and the region (r1 , r2 ) is excluded for T1 (t). In particular if T1 (t0 ) ≤ r1 , (6.185) T1 (t) ≤ r1 , (6.186) then for ∆t satisfying (6.175). In particular T1 (t) ≤ C| ln | T1 (t0 ) + β , (6.187) which in terms of W , for ∆t exactly satisfying (6.175), becomes ||W (t)||X ≤ C ωβ ||W (t0 )||X + β , (6.188) for any 0 < ω < 1 and any t ∈ (t0 , t0 + ∆t), where the | ln | term is absorbed in the ωβ term by taking ω slightly smaller. 154 6.3 The renormalization group iteration We can now iterate the estimates above to an equilibrium, much as in the application of renormalization group methods to statistical mechanics. We break our evolution equation into a series of initial value problems. We fix ω < 1 and define the renormalization times as tn = tn−1 + ∆t, where ∆t is given in (6.175). On the disjoint intervals In = [tn−1 , tn ], we have initial data W (tn ) ∈ Xp , with T1,n and T2,n n corresponding to (6.39) and (6.40). The renormalization group map G takes the initial data Wn−1 = W (tn−1 ) for the initial value problem on In−1 and pulse positions pn−1 = p(tn−1 ) and returns the initial data Wn = W (tn ) and pulse positions pn = p(tn ) for the next initial value problem:      Wn−1   Wn  G . = p(tn ) p(tn−1 ) (6.189) This map includes both the evolution under the flow, and a reprojection under Proposition 6.1. The initial data and the new base point pn are obtained from W (t− ), n the right end point of the evolution of W over In−1 . This process is illustrated in Figure 1.1. To bound the renormalization group map, we must control the secular jump under 155 the projection. From Lemma 6.1, pn = p(t− ) + H W (t− ) . n n (6.190) Since W (t− ) ∈ Xpn−1 , we apply the estimate (6.3), which becomes n |pn − p(t− )| ≤M1 ||W (t− )||X T2 (∆t) n n ≤M1 ||W (t− )||X n − β + ||W (t− )|| n X ||W (tn )||X , (6.191) (6.192) where we applied estimates on T2 from (6.164) and ∆t from (6.175). We decompose the solution at time tn as U (tn ) = Φ p(t− ) n + W (t− ) = Φp(t ) + W (tn ). n n (6.193) We can bound the jump of W at each renormalization as ||W (tn ) − W (t− )||X =||Φ n − Φp(t ) ||X n p(t− ) n ≤C|p(t− ) − p(tn )| n (6.194) (6.195) ≤M1 − β + −α T 2 1,n−1 ||W (tn )||X (6.196) ≤M1 − β + ||W (t− )|| n X ||W (tn )||X , (6.197) 156 2 since Φ is X-Lipschitz in p. Since −α T1 1, the renormalization step is asymp- totically negligible and we recover the estimate |Wn ||X ≤ M ωβ ||Wn−1 ||X + β . (6.198) To control ||W ||X on the long, renormalization group time scale, we may introduce ηn+1 =M ωβ ηn + β (6.199) η0 =||W0 ||X , (6.200) so that ||Wn ||X ≤ ηn . (6.201) M (1+ω)β , ωβ M 1− (6.202) It is easy to verify that ηn → as n → ∞ for η0 small enough. The overall evolution for W may be written as ||W ||X ≤ M α − 2 µt e ||W0 ||X + (1+ω)β for any 0 < ω < 1. 157 , (6.203) 6.4 Long-time asymptotics After the residual has relaxed into its O (1+ω)β equilibrium layer about the manifold Φ, the pulse dynamics in (6.114) reduce to pj = − ˙ R2 (Φ) + [∆L(W + Φ1 )]2 + N2 (Φ1 , W ), φj 2 L +O −2α21 /(α22 −1) qj ||φ0 ||2 2 L 2 . (6.204) Using (6.175), (6.183), and (6.187) to estimate (6.164), we have 2−β ∆t + 2 + −α T (t)2 1 T2 (t) ≤C1 ≤C β + 2 + −α | ln | T (t ) + β 1 0 ≤C β , if (6.205) 2 (6.206) (6.207) 1+α 2 4 T1 (t0 ) ≤ . | ln( )| 158 (6.208) The above it is more exigent then (6.183), so it is the constraint on ||W0 ||X in Theorem 1.1. We use (6.43), (6.46), and (6.207) to bound the following: ˜ ||[∆LW ]2 || 2 ≤C L ≤C 2−β + T (t) 2 ||W ||X + 2−β 2 + β ||W || X (6.209) (6.210) ≤C 2 . (6.211) Also for the nonlinearity, using (6.48) we have ˜ ||N2 (W )|| 2 ≤ C ||W ||2 + 4−2β ≤ C 4−2β . X L (6.212) So the evolution equation reduces (6.204) to pj = − ˙ R2 (Φ), φj 2 L +O −2α21 /(α22 −1) 2 qj ||φ0 || 2 L 159 2 . (6.213) Taylor expanding, we can reduce R2 (Φ), φj 2 to L  R2 (Φ), φj 2 =  L =  N α α α Φ1 21 − qi 21 φi 22 , φj  i=1 L2 α α +O Φ1 21 (pj )(x − pj )φj 22 , φj L2 (6.214) 2 (6.215) α α −1 =α21 Φ1 21 (pj )Φ1 (pj ) (x − pj )φj 22 , φj 2 + O L 2 (6.216) α −1 α =α21 qj 21 Φ1 (pj ) (x − pj )φj 22 , φj 2 + O L 2 . (6.217) Continuing, we integrate by parts on the above inner product where α −1 α R2 (Φ), φj 2 =α21 qj 21 Φ1 (pj ) (x − pj )φj 22 , φj 2 + O 2 L   L α22 +1 φj α21 −1   2 = − α21 qj Φ1 (pj ) 1,  +O α22 + 1 L2 α21 α21 −1 α +1 qj Φ1 (pj )φj 22 +O 2 . =− α22 + 1 160 (6.218) (6.219) (6.220) Substituting the definition φj (x) = φ0 (x − pj ) from (3.4), the above expands to α21 /(α22 −1) qj −α21 α21 −1 α +1 R2 (Φ), φj 2 = qj Φ1 (pj )φj 22 +O 2 α22 + 1 L α −1 qj 21 Φ1 (pj )α21 α +1 +O φ0 22 =− α22 + 1 α21 (α22 +1)/(α22 −1) qj (6.221) 2 α21 (α22 +1) Φ1 (pj )α21 α21 −1− α −1 α +1 22 =− qj ||φ0 || 22 +1 + O α22 + 1 Lα22 (6.222) 2 . (6.223) Now returning to (6.213), and substituting the above, we have R2 (Φ), φj 2 L +O 2 −2α21 /(α22 −1) qj ||φ0 ||2 2 L α22 +1 α21 (α22 +1) 2α21 Φ1 (pj )α21 ||φ0 || α22 +1 α21 −1− α −1 + α −1 L 22 22 +O = qj 2 α22 + 1 ||φ0 || 2 L α22 +1 ||φ0 || α +1 α21 1 L 22 = Φ1 (pj ) + O 2 , 2 α22 + 1 ||φ || qj 0 L2 pj = − ˙ 161 (6.224) 2 (6.225) (6.226) where α (α + 1) −α21 α22 − α21 + 2α21 α21 − 1 − 21 22 + 2α21 /(α22 − 1) =α21 − 1 + α22 − 1 α22 − 1 (6.227) =α21 − 1 + −α21 (α22 − 1) α22 − 1 = − 1. (6.228) (6.229) Then we can express the dynamics of the pulse position as α +1 ||φ0 || 22 +1 α21 1 Lα22 pj = ˙ Φ (p ) + O α22 + 1 ||φ ||2 qj 1 j 0 L2 2 . (6.230) We know from (3.44), that Φ1 (pj ) is O( 2−β ) in L∞ , so the above will generate leading order dynamics. We want to determine a representation for Φ1 (pj ). We substitute the definition of φj (x) from (3.4) into the definition of Φ1 in (3.2), where  Φ1 = −β L−e  11  = −β L−e  11 N j=1 N  α α qj 11 φj 12 (x) (6.231)  α −α α /(α22 −1) α12 φ0 (x − pj ) qj 11 21 12 j=1 162 (6.232) Differentiating the above, we have  Φ1 = −β α12 L−e  11 N  θ α −1 qj φ0 12 (x − pj )φ0 (x − pj ) , (6.233) j=1 for θ defined in (2.67). Then substituting the above, we expand term Φ1 (pj ) in (6.230) as Φ1 (pj ) = Φ1 (x), δpj   N θ α −1 qk φ0 12 φ0 , L−e δpj  = −β α12  11 k=1   N θ α −1 qk φ0 12 φ0 , G0 (x − pj ) = −β α12  k=1   N 1+α/2 √µ|x−p | π − θ α −1 j , qk φ0 12 φ0 , =α12  e 2µ k=1 where G0 (x) = (6.234) (6.235) (6.236) (6.237) π 1−α/2 − 1+α/2 √µ|x| e , the Green’s function defined in (2.49). 2µ We also used the fact that 1−α/2−β = 0. Then continuing the above and integrating 163 by parts, we have  N Φ1 (pj ) =α12  = = θ α −1 qk φ0 12 φ0 ,  1+α/2 √µ|x−p | π − j  e 2µ (6.238) k=1   N 1+α/2 √µ|x−p | π  − θ α j  qk φ0 12 (x − pk ) , e (6.239) 2µ k=1 N √ π 1+α/2 α − 1+α/2 µ|x−pj | θ qk φ0 12 (x − pk ), sign(x − pj )e 2 k=1 (6.240) = +O = +O π 2−β 2 N θ qk √ α12 − 1+α/2 µ|pk −pj | φ0 (x − pk ), sign(pk − pj )e k=1 4−2β (6.241) π 2−β α12 φ0 2 N θ qk √ − 1+α/2 µ|pk −pj | sign(pk − pj )e k=1 4−2β . (6.242) Combining this with (6.230) in vector form yields the following representation of the pulse dynamics: ˙ p = 2−β Qθ A(p)q −1 + O 164 2 ), (6.243) where Q is the diagonal matrix of the amplitudes q and the antisymmetric matrix A(p) is defined componentwise as              Akj =          −    α +1 ||φ0 || 22 +1 1+α/2 √µ|p −p | π α21 Lα22 φα12 e− k j 0 2 α22 +1 ||φ ||2 0 L2       k>j        0 k = j .   α +1   √  ||φ0 || 22 +1 1+α/2 µ|p −p |  α22 π α21 α12 −  L k j k 0, where the smaller interval is called the nose of the essential spectrum. One would need to introduce two different spectral projections, developed as appropriate contour-deformation limits of the resolvent. The semigroup estimates would also be established as a limit in which the contour of integration relaxes onto the nose the essential spectrum. A key difficulty in this analysis would be projecting the essential spectrum off of the nose and assuring that the spectral projection faithfully captures any embedded eigenvalues, uniformly as → 0+ . The second extension would examine a more general nonlinearity. With a general nonlinearity, the problem could encompass many equations including the regularized Gierer-Meinhardt equation and the Gray-Scott equation. The goal would be to unify previous research and new results under one inclusive work. This would offer a general system that could be used for many specific reaction-diffusion equations. One could also also pursue the problem in the case of multiple activator components to achieve similar results. The third extension would be to prove the existence of radial spot solutions for a general equation and study their stability properties. One could also examine the interaction of multiple spot patterns. Here U and V are functions of two space 167 dimensions. Previous results have formally been extended to the two-dimensional spatial spot problems, including [8], [16], [17], [27], [33], [35], [36], and [37]. Many of these works include the Gierer-Meinhardt or the Gray-Scott model. In the future, we hope to rigorously extend similar results to the two-dimensional setting. 168 BIBLIOGRAPHY 169 BIBLIOGRAPHY [1] P. W. Atkins and J. De Paula, Physical Chemistry, 7th edition, Oxford University Press, Oxford, (2002). [2] P. W. Bates and C. K. R. T. Jones, Invariant Manifolds for Semilinear Partial Differential Equations, Dynam. Report., 2, (1989), 1-38. [3] P. W. Bates, K. Lu, and C. Zeng, Existence and persistence of invariant manifolds for semiflows in banach space, Mem. Amer. Math. Soc., 135 (645), (1998). [4] J. Bona, K. Promislow, and C. Wayne, Higher order asymptotics of decay for nonlinear, dispersive, dissipative wave equations, Nonlinearity, 8 (6), (1995), 1179-1206. [5] J. Bricmont and A. Kupiainen, Renormalizing Partial Differential Equations, Constructive Physics, Lecture Notes in Phys., 446, (1995), 83-115. [6] J. Bricmont, A. Kupiainen, G. Lin, Renormalization group and asymptotics of solutions of nonlinear parabolic equations, Comm. Pure. Appl. Math., 47, (1994), 893- 922. [7] L-Y. Chen, N. Goldenfeld, and Y. Oono, Renormalization group theory for global asymptotic analysis, Phys. Rev. Lett., 73, (1994), 1311-1315 . [8] W. Chen and M. Ward, The Stability and Dynamics of Localized Spot Patterns in the Two-Dimensional Gray-Scott Model, SIAM J. on Appl. Dyn. Sys., (2011), to appear. [9] A. Doelman, W. Eckhaus, and T. Kaper, Slowly-modulated two-pulse solutions in the Gray-Scott model I: Asymptotic construction and stability, SIAM J. on Appl. Math, 61 (3), (2001), 1080–1102. [10] A. Doelman, R. A. Gardner, and T. Kaper, Large stable pulse solutions in reaction-diffusion equations, Indiana Univ. Math J., 50 (1), (2001), 443–507. 170 [11] A. Doelman and T. Kaper, Semi-strong pulse interactions in a class of coupled reaction-diffusion equations, SIAM J. on Appl. Dyn. Sys., 2, (2003), 53–96. [12] A. Doelman, T. Kaper, and K. Promislow, Nonlinear asymptotic stability of the semistrong pulse dynamics in a regularized Gierer-Meinhardt model, SIAM J. Math. Anal., 38 (6), (2007), 1760-1787. [13] A. Doelman, T. Kaper, and K. Promislow, Hysteric behavior in semi-strong pulse interactions in the gray-scott model, preprint. [14] Shin-Ichiro Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J.D.D.E., 14 (1), (2002), 85-137. [15] S.-I. Ei, M. Mimura, and M. Nagayama, Pulse-pulse interaction in reactiondiffusion systems, Physica D, 165 (1), (2002), 176-198. [16] S.-I. Ei, M. Mimura, and M. Nagayama, Interacting Spots in reaction diffusion systems, DCDS, 14 (1), (2006), 31-62. [17] S.-I. Ei and J. Wei, Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimension, Japan J. Ind. Appl. Math., 19 (2), (2002), 181-226. [18] A. Gierer and W. Meinhardt, Theory of biological pattern formation, Kybernetik, 12, (1972), 30-39. [19] N. Goldenfeld, Lectures on phase transitions and the renormalization group, Frontiers in Physics, Vol. 85, Addison-Wesley, Reading, MA, (1992). [20] N. Goldenfeld, O. Martin, and Y. Oono, Intermediate asymptotics and renormalization group theory, J Scientific Comput, 4 (4), (1989), 355-372. [21] N. Goldenfeld, O. Martin, Y. Oono, and F. Liu, Anomalous dimensions and the renormalization group in a nonlinear diffusion process, Phys. Rev. Lett, 64, (1990), 1361-1364. 171 [22] P. Gray and S.K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor - oscillations and instabilities in the system A + 2B → 3B; B → C, Chem. Engineering Science, 39, (1984), 1087-1097. [23] M. Guha and K. Promislow, Front propagation in a noisy, nonsmooth excitable media, Disc. Cont. Dyn. Sys., 23 (3), 617-638. [24] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, (1981). [25] H¨rmander, L, The Analysis of Linear Partial Differential Operators 14. o Springer, New York, (1985) [26] D. Iron, M. J. Ward, and J. Wei, The stability of spike solutions of the onedimensional Gierer-Meinhardt model, Phys. D, 150, (2000), 25-62. [27] T. Kolokolnikov and M. J. Ward, Reduced wave Green’s functions and their effect on the dynamics of a spike for the Gierer-Meinhardt model, European J. Appl. Math, 14, (2003), 513-545. [28] R. O. Moore and K. Promislow, The semistrong limit of multipulse interaction in a thermally driven optical system, J. Diff. Eq., 245 (6), (2008), 1616-1655. [29] K. Promislow, A renormalization method for modulational stability of quasisteady patterns in dispersive systems, SIAM J. Math. Anal., 33 (6), (2002), 1455-1482. [30] B. Sandstede, Stability of travelling waves, Handbook of Dynamical Systems, II, B. Fiedler, ed., North-Holland, Amsterdam, (2002), 983-1055. [31] A.M. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc., 237, (1952), 37-72. [32] P. van Heijster, A. Doelman, T. Kaper, and K. Promislow, Front interactions in a three-component system, SIAM J. on Appl. Dyn. Sys., 9, (2010), 292-332. 172 [33] P. van Heijster, B. Sandstede, Planar radial spots in a three-component FitzHugh-Nagumo system, J. Nonlinear Sci., (2011), to appear. [34] M. J. Ward and J. Wei, Hopf bifurcation of spike solutions for the shadow GiererMeinhardt model, European J. Appl. Math, 14, (2003), 677-711. [35] J. Wei, On single interior spike solutions of the Gierer-Meinhardt system: uniqueness and spectrum estimates, European J. Appl. Math, 10, (2003), 353378. [36] J. Wei, Pattern formations in two-dimensional Gray-Scott model: existence of single-spot solutions and their stability, Phys. D, 148, (2001), 20-48. [37] J. Wei and M. Winter, Asymmetric spotty patterns for the Gray-Scott model in R2 , Stud. Appl. Math., 110, (2003), 63-102. 173