PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 2/05 p:ICIRC/Dale0ue.6ndd-p.1 TRAVELING PULSES FOR THE NONLOCAL AND LATTICE KLEIN-GORDON EQUATIONS By Chunlei Zhang A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 2006 ABSTRACT TRAVELING PULSES FOR THE NONLOCAL AND LATTICE KLEIN-GORDON EQUATIONS By Chunlei Zhang The thesis includes two parts. In the first part, we study a nonlinear nonlocal Klein-Gordon equation on the whole real line. We first prove the existence of traveling pulses and then study the spatial asymptotic properties of these pulses and their instability. In the second part, we study a nonlinear lattice Klein-Gordon equation, which is actually a system of infinitely many ordinary differential equations. We show the existence and determine the spatial asymptotic properties of the traveling pulses using a similar method to that in the first part and then study the instability by numerical simulations. For both the nonlinear nonlocal and nonlinear lattice Klein- Gordon equations we illustrate that they share some similar properties with the limit equation, the classical Klein-Gordon equation, as a parameter 5 approaches 0. To my family iii ACKNOWLEDGMENTS I would like to thank my dissertation advisor, Professor Peter W. Bates, for his patient guidance, and for his encouragement and support during my graduate study at Michigan State University and Brigham Young University. He helped me to learn not only how to do research in mathematics, but also to write mathematical papers using proper English. The education I received under his guidance will be most valuable in my future career. I would also like to thank Professor Kening Lu and Professor Tiancheng Ouyang for their advice when I studied at Brigham Young University. Many thanks to my dissertation committee members Professor Keith Promislow, Professor Guowei Wei, Professor Baisheng Yan, and Professor Zhengfang Zhou for their time and advice. Finally, I especially thank my wife, Wenmei, and my parents for their encourage- ment and support. iv TABLE OF CONTENTS LIST OF FIGURES 1 Introduction 2 Traveling pulses for the nonlocal Klein-Gordon equation 2.1 Existence of traveling pulses ..................... 2.2 Proof of the Theorem ......................... 2.3 Asymptotic behavior of traveling pulses ............... 2.4 Instability of traveling pulses ..................... 3 Traveling pulses for the lattice Klein-Gordon equation 3.1 Existence of traveling pulses ..................... 3.2 Asymptotic behavior of traveling pulses ............... 3.3 Instability of traveling pulses ..................... BIBLIOGRAPHY vi 10 10 14 22 24 3O 3O 33 36 47 1.1 1.2 1.3 1.4 1.5 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 LIST OF FIGURES vertical displacement ............................. 2 f(u) ....................................... 7 phase portrait ................................. 8 single pulse .................................. 8 traveling pulse ................................. 9 un(t),t = .................................. 38 un(t),t — 5 .................................. 38 un(t),t = 6 .................................. 39 un(t),t = .................................. 39 un(t),t = 8 .................................. 40 un(t),t = 10 .................................. 40 un(t),t = 15 .................................. 41 un(t),t = 20 .................................. 41 un(t) - (1)0(511 — ct), t=1 ........................... 42 un(t) - ¢0(5n — ct,) t = 5 ........................... 42 un(t) — ¢0(5n - Ct,) t“ — 6 ........................... 43 un(t)- 430(511 - ct), t— — 7 ........................... 43 un(t) — ¢o(en - ct), t- — 8 ........................... 44 un(t) — (150(572 — ct), t - 10 .......................... 44 un(t) - ¢o(€‘n — Ct),t = 15 .......................... 45 un(t)— ¢0—'(En ct),t = 20 .......................... 45 vi CHAPTER 1 Introduction Crystals are solids in which the constituent atoms are arranged regularly in a space lattice with specific geometrical symmetry elements. However, a perfect crystal, with every atom of the same type in the correct position, does not exist in nature. Actually, all crystals have defects. Crystal defects are very important in materials science and engineering because they govern many physical properties, such as color, conductivity, transparency, hardness, etc.. There is also much work on the mathematical model studying crystal defects (see [22] for instance). Kresse and Truskinovsky [23] studied a lattice wave equation with nearest neighbor interaction to study the motion of lattice defects (see [34] for an analysis with longer range interaction). They consider a chain of particles attached at equal distance 5 to a rigid background by bistable springs with energy density w(u,,) where an is the vertical displacement of the particle with index 11 (Figure 1.1), then sw(u,,) gives the on site potential energy for particle with index n. Assume that the neighboring particles interact through standard harmonic forces, characterized by the modulus of elasticity 6 (Young’ modulus) of the spring, then the total potential energy of the chain is in the form —- o———o———+————c————o~—r~—A -—-———r————+ <1 2 1 o 1 2 a x O O O O o . L A A .—-’ -3 -2 1 1 a x Figure 1.1. vertical displacement 1 un — un V = Z: §£€(—+IE‘—)2 + 510(11"). For simplicity assuming the spring has unit mass density, then the total kinetic energy is K = Z 15.22. n 2 Then the Euler-Lagrange equation gives 8 11,, — E§[un+1 — 211,. + un_1] + w'(un) = 0. The authors consider traveling waves, and so let :1: = an, write un(t) = u(2:, t) and seek a solution of the form u(:r, t) = u(77) = u(:z: — ct) satisfying the equation c2561; — ghm + e) — 211(1)) + u(n — 5)] + w'(u) = 0. (1.1) In [23] and [34], heteroclinic solutions were studied for a piecewise linear w’, in which case solutions can be found explicitly. We consider a general lattice equation .. 1 °° u, — 25 Z akun_k + f(un) = o, n e z, (1.2) kz-oo where O < e, Z 0;. = 0, a0 < 0, 0,, = a4, and a certain ellipticity condition holds. Note that we allow infinite range and not just nearest neighbor or finite length interaction, although those are included as special cases. We want to study homoclinic waves for supersonic speeds c and rather general smooth nonlinearity. We assume that f is a 02 function, satisfying f (O) = 0, and Co =inf{C >0=F(C) =0} exists, where no = [C f(8)ds. and f ((0) > 0. A typical example is the quadratic function f (u) = u(u — a) with a > 0. Under such assumption the ODE u" + f (u) = 0 has a unique even, positive homoclinic solution (see [8]). Equation (1.2) can be derived through the Hamiltonian. For instance, one could take the Hamiltonian on [2 x [2 defined by __ 1 2 1 2 H(pv 11) _ Z(§pn + E gun—main - um) + F(un)) (13) 71 Here, p = (pn) E 52 represents the momentum and u = (an) E [2, the position of particles relative to lattice sites. The pairwise interaction term, ()z,,_,,,(un —um)2, could be thought of as each pair of particles being connected by a linear spring. However, we will not require that these all produce restoring forces, i.e., some of the azs could be negative. Such is the case with the Lennard-Jones potential, for instance. Since the term with n = m vanishes, we may take 00 = — 2h“) ak. We will also assume interactions are symmetric, so that a_k == ak. 3 The on-site potential F is fairly general but has a local maximum or semi- maximum at 0 and a local minimum at some positive value, 1 say. Examples include F (u) = (1 — 14"")2 or u2(u — 1), which cases have been well studied numerically, and analytically in the continuum differential equation setting. The equations of motion produced by (1.3) are dun __ 6H _ dt — 6p ,, _ pm dpn 6H 1 a = ’ (a) = 252m a._...u.. - “W, where f = F’. This system is equivalent to (1.2). Note that if we rescale n in (1.1) such that z = 77/5, the equation for u(z) = u(sn) becomes 2dzu c d—zz- — £[u(z +1) — 2u(z) + u(z —1)]+ e2w'(u) == 0 and many authors have studied such lattice Klein-Gordon equations (see [9, 14, 16, 17, 30], and the references therein). Some computational studies on heteroclinic traveling waves for a cubic bistable 111’ have been reported in [9], and the existence of homoclinic traveling wave solutions for fixed 5 > 0 and sufficiently small 8 has been proved in [20] by center manifold reduction (see also [14]). The existence of other solitary solutions, for example, breathers, was studied in [24, 31]. However, for equation (1.1), when w’ is nonlinear, s is small, and 8 is of moderate size, little is known about the existence of traveling waves or pulses. A major difficulty arises as one must deal with a nonlinear differential equation with significant advanced and retarded terms. That case is considered here. Many other nonlinear infinite dimensional lattice systems have been studied during the last couple of decades. For example, the famous FPU lattice, fin = V'(un+1 — u") — V'(un — un_1) has received considerable attention, in particular having been studied by numerical simulations [13] (see also [12, 15]). More recently there have also appeared some rigorous results concerning traveling waves using a variety of methods. For instance, there are variational approaches used in [19, 26, 33]. In [18], Friesecke and Pego studied the qualitative properties of traveling pulses in F PU lattices from another point of view: they regard the traveling pulse as a perturbation of the solitary solution of a KdV equation. Their method is to look for the fixed point of an operator defined in terms of Fourier multipliers, but which is a reformulation of the problem at hand. The Fourier transform has also been used by other authors to study solitary waves (see [1, 12] etc, for both numerical and theoretical treatments). In this work, we will prove our main result following the idea of [18]. There is also some literature considering traveling waves for the damped discrete sine-Gordon equation with periodic boundary conditions (so it is a finite-dimensional system rather than an infinite-dimensional one), which models many physical systems such as circular arrays of Josephson junctions. The existence of traveling waves was proved in [21] by a fixed-point argument. At present, our methods do not apply to such systems, however. Note that in (1.2) we can regard the term 1 oo Aeun E :2 Z akun—k kz-oo as a discrete Laplacian with possibly infinite range interaction. Similar long range interactions have been introduced in a DNA model with long—range dipole-dipole interactions. The typical Hamiltonian is of the form N N 1 H: E (4.2 +V(u,,)+ E Jmnunum) m=l 2 71 11:1 and traveling breathers were numerically studied in [2, 3, 11]. While we do not intend for the foregoing to be a complete survey of previous results on lattice equations and traveling waves, we hope that it provides a context for our results. We also consider the nonlocal wave equation for u(2:, t): 1 . utt-;§(35*u-u)+f(u)=0, for t>0 and xelR, (1.5) where 5 is a positive parameter and the kernel jE of the convolution is defined by where j() is an even function with unit integral. We can regard equation (1.5) (or (1.2) ) as the wave analog of the nonlocal parabolic equation studied in [4], (or [5] for the lattice version) Mme—1.0” - u) +f(U) -—- o, (1.6) (see also [6, 7, 10]). Equation (1.6) was introduced as an L2 gradient flow for the energy functional 1 . E = 3 f [mm - y)>2dxdy + [R F>dx. Similarly, when we apply Newton’s second law at, = —grad E(u), a nonlocal wave equation, (1.5), is derived. 1 . Note that, as 5 -> O , E—2(je * u — u) —> dun, formally and in some weak sense described in [4], where d is a constant determined by j. So we can also regard (1.5) as a nonlocal version of the classical nonlinear Klein-Grodon equation u“ — u... + f(u) = 0. (1.7) Figure 1.2. f(u) Clearly, a traveling wave solution 21(7)) = u(x - ct) satisfies the ordinary differential equation (02 —1)u" + f(u) = 0 which has homoclinic solutions for speed c2 > 1 and suitable nonlinear function f. For example, with f = u3 — u (Figure 1.2), the phase portrait is shown in Figure 1.3 and the solution is a single pulse (Figure 1.4). Actually, the ODE is integrable and the solution is given by a sech function u(n) = \/23ech( 7) ) . 02 — 1 Figure 1.5 shows how the pulse travels when the speed c is positive. In this work we will study homoclinic traveling wave solutions (traveling pusles) of (1.5) (and of (1.2)), i.e., solutions of the form u(x, t) = u(:r — ct) (or un(t) = u(en — ct))which decay to zero at infinity. It will become apparent that for both the lattice and continuum equations, our method does not seem to apply to the subsonic case c2 < d. Also, the Lorentz invariance, which produces, for stationary solutions, subsonic traveling waves for the Figure 1.3. phase portrait Figure 1.4. single pulse Figure 1.5. traveling pulse usual nonlinear wave equation, does not hold for the nonlocal version. Since the corresponding limit ODEs have heteroclinic solutions for bistable nonlinearity f, one may conjecture that there exist heteroclinic traveling waves for our models, at least when a > 0 is small. On the other hand, even though the limit equation does not have homoclinic solutions, since the convolution operator is bounded, we suspect that our models still possess traveling pulse solutions with subsonic speed provided 6 is sufficiently large. The dissertation is organized as follows: In chapter 2 we study the traveling pulses for the nonlocal Klein-Gordon equation (1.5). We first show the existence and spatial asymptotic properties of the traveling pulses, then we prove that the traveling pulses are unstable. In chapter 3 we study the lattice Klein-Gordon equation (1.2). We show the existence of pulses, and determine their spatial asymptotic properties using similar methods to those in Chapter 2, then illustrate the instability thorough numerical simulations. CHAPTER 2 Traveling pulses for the nonlocal Klein-Gordon equation 2.1 Existence of traveling pulses In this section we study the existence of traveling pulses for the nonlocal Klein—Gordon equation (1.5). Throughout this dissertation we assume the following conditions: (A1) f E 02(R),f(0) = 0, NW = -a < 0; f((o) > 0, where C (0 _=. inf{( > O : F(() = 0} and F(C) 2/0‘ f(s)ds, and (A2) j(.r) 6 DUI?) is even, has unit integral, ’> _ 2 Ho 22 and 3(2) 2 1 (112 , where 0 < d 3 (11 are constants and the Fourier transform is given by 3(2) 5 / e"”j(x)d:r. ---00 Remark A typical example for f is the quadratic function f (u) = u(u — a) with 10 a > 0. The assumption (A1) guarantees that the ODE u" + f(u) = 0 has a unique even, positive homoclinic solution ([8]). A 1 °° 1 Remark Ifj E C2 then d = 5/ j(x)x2da:. It is easy to see that -2-e"’| , -oo 1 2 l l l fire—$2 and Til—4:272- satisfy (A2), with d(= d1) given by 1, 4 and 5 respectively. The idea is to change to traveling wave coordinates, formally take the Fourier transform, and reformulate the problem of existence of a traveling pulse as a fixed point problem for an operator defined in terms of the nonlinearity and a certain Fourier multiplier. In (1.5), let 21(33, t) 2 11(2: - ct) = 11(7)), so that 11(0) should satisfy the equation 1 . c221” — 5—205 * u — u) = —g(u) + au, (2.1) where g(u) = f (u) + an. Applying the Fourier transform, equation (2.1) becomes 1 ’.‘ A —c2€2f1 — E505 - f1 — f1) = —g(u) + air 01' A (0252 +11(6) + (1)22 = 9(a), A l where L; 2 E595 - 1). Thus, an equivalent formulation is A 21 = 1940901). where (5) - 1 <2 2) P5 — c2€2+le(£)+a' ° The inverse Fourier transform gives u = 155 * 9(U), (2-3) 11 where 15,. is the inverse transform of 1),. Define the operator P411); 13. * 9(a), and write (2.3) as u = P5(u). (2.4) Note that, due to (A2), A j(€€)-1 .2 __ 2 (as)? ‘5" d5 :45) 5—12-(2'10- 1) = as e —> 0, so p, z (c2 _- (11)? +a for 5 small. Thus, when 5 —> 0, (2.4) formally becomes u = POOL), (25) where P0(u) E 150 * g(u) and 1 p0(€) '— (C2 _ d)€2+a° Clearly, (2.5) is equivalent to u = ((d - 62W + a)’1(9(U)), that is, (C2 - d)U" = an - 901), or (c2 — (1)21" + f(u) = 0. (2.6) By the results of Berestycki and Lions in [8], under the assumption (A1), (2.6) has a unique even, positive homoclinic solution for each c2 > d, which we denote by 110. Thus, 110 is a fixed point of operator Po. We can write equation (2.5) in the form U = POW) + (P: - FOX"), 12 and look for a fixed point near uo. The basic idea to solve this is to consider 21 = u—uo. Since '11 satisfies fi=P5(’l-l.—UO) —P0(U), which is equivalent to 1'1 — DP0(uo)r1 = P501 — uo) — Po(u) — DP0(uo)u, we want to solve the equation 1‘1 = (I — DPo(uo))’l(Pe(u + uo) — Po(uo) — DPo(uo)1‘1). (2.7) The operator on the right hand side turns out to be a contraction map on a small ball about 0 when e > 0 is sufficiently small. Under these conditions the operators F = P0 and F = P5 — P0 satisfy the assumptions in the following lemma from [18] : Lemma 2.1.1 Suppose F and F are C1 maps from a ball in a Banach space E into E. Suppose F has a fixed point (to and that A E I — DF(¢0) is invertible with [IA‘III S Co < 00. Suppose that there exist positive constants Ci,Cz,0, and 6 such that (i) IIDF(¢) - DF(¢o)II s C] < 05“ and IIDF(¢)H s 02 whenever ”45 " ¢0||E S 5, where I] ' I] represents the operator norm ; (ii) 00(01-1“ 02) S 0 <1; (1'11) [IFWolliE < 5(1- 0)/Co. Then the equation 45 = F(¢) + F(¢) has a unique solution satisfying ||q5 — 450”); S 6. This will be used to prove our main theorem: 13 Theorem 2.1.2 Under the assumptions (Al) and (A2), for any speed c satisfying c2 > d1 there exits an 50 > 0 such that for any 5 6 (0,80), equation (1.5) has a unique nonzero solution us in the set {u E H1(R): u is even, [[11 — 210””: < 6}, where uo is the even, positive homoclinic solution of (2.6) and 6 > 0 depends on j, f and c, and satisfies 6 < ”Hollyl. Remark The condition 6 < ”UoliHl excludes the trivial solution u _=_ O. 2.2 Proof of the Theorem Note that uo is the fixed point of P0. To prove the theorem we apply Lemma 2.1.1 with the operators F=P.—P0, and F=Po. First, we show F and F are 01 maps in E = H; = {u E H1(R) :u is even}. Step 1 PE is an operator from H 1 to H 1 . Denote [I ~ [I = II - [le and [I - ”00 = I] ' ”Loo. We also use [I - I] for the operator norm, where the range and domain spaces are understood. For any u 6 H‘, by Plancherel’s theorem, IIPe(U)|| = Ilfis * 9(U)II 1 A = -—|lpeg(U)|l V577 (2.8) S “peiloo”9(u)“ S llpelloollg(U)/UI|oollall. 14 where g(u)/u E L°° because g’(0) = 0 and u E L°°. We also have, ]|(P5(u))’|I = “155 * (9'(U)U')|| = -—l—up.(gTuTu')u v27r S. ”FEHOOHQIWWIH S IIPelloollg'(U)llooIIU'||- Thus, PE is an operator from H1 to H1. Step 2 P,E maps even functions to even functions. Let u be even, then g(u) is even and it is easy to check that the convolution of two even functions is even. Step 3 For u E H1, the linearized operator DP5(u) : H1 —> H l is bounded. For any v 6 11‘, Thus, IIDPe(U)(v)|| = “15: * (9’(U)v)ll S llpEIIOOHg’(u)|I00”v”a and ||(DP.(U)(v))'|I = ”(15. * (9’(U)v))’|| S llpsllooll(9'(U)v)’ll S Ilpelloo(||9”(U)llooIIU’llllvlloo + ||9’(U)lloo||v’ll) S Cllvllm, where we used the embedding inequality, “'UHLOO S C]|v|| H1. Therefore, DP€(21) is bounded. 15 Step 4 The operator DP€(u) is continuous with respect to u. For any 11,211 6 H1, llDPe(U)(v)-DP5(UI)(U )ll =llps* [(g’m )- 9’(U1))vlll S llpelloollg’W) - 9’(ul)|loollv|| S llpelloollg”lloo|lu - uilloollvll, and ||(DP.(u)(v) - DPe(u1)(v))’ll = ”15. * [(9’(U) - 9’(ui))v]'ll = H13. * [(g”(U)u’ - 9”(U)U'1 + g”(u)U’1 - 9”(u1)U’1)vl+ 15. * [(9’(U) - 9’(ul))v’lll S llpelloollg"(u)llooIIU’ - u’lllllvlloo + “pelloollg”(u) - 9"(ui)l|ooIIU’1lloo||v|| +Hpell00llg’(u) - g'(ui)|loollv’||- Hence IIDPAU) - DRUM)“ -+ 0 as ||u — all] Hi —> 0. Therefore DPS is continuous and P,E is 0‘. Similarly, P0 is a C 1 map from E to E. We will finish the proof using the following lemmas. Lemma 2.2.1 The operator T = DF(u0) : H1(lR) —> H1(IR) is compact. Proof: Note that DF(Uo)v = 150 * (9'(Uo)v) l (c2 - d)€2 + a 1 = W — 1)./16‘5““ (gl(i‘°)”)’ = ( )'* (9,010)”) l6 / a , . where 6 = 2 d. Since uo decays at 21:00 , we can prove that DF(u0) 1s compact c —. using the Fréchet-Kolmogorov Theorem (see [35]): A subset K in LP (1 S p < 00) is precompact if and only if it satisfies 1- SUP ”UHF < 00; 116K 2. ||u(- + h) - u(-)||,, —) O uniformly in u E K as h —> 0; 3. lim |u(a:)[”d:t: = O uniformly in u E K. 0—)“) '1')“ First we show T : L2 —+ L2 is compact. Let A be a bounded subset in L2 and let K = TA. We need to show K is precompact in L2. (1) As in (2.8) we have [[Tv|]2 g C(u0)||v||2 for v E A. (2) For v E A, foo |Tv(a: + h) - Tv(x)|2d:r = ll(fio(- + h) - 150(0) * (9'(Uo)v)||§ S “(150(- + h)T- Pot)||§.||9’(uo)l|§ollv|l3 -+ 0 uniformly in v 6 A as h -> 0 because [(150(' + fill“) — 19005)] = leihépogl - Poigli = le‘“ -1lpo(€) -> 0 uniformly in E as h ——> 0, since p0(§) ——) 0 as [6] —-) oo. 3) Since no decays at ice, for any 7 > 0, there exists M > 0 such that |9’(Uo(y)l<7 for |y|>M. 17 Choose a > M such that 2M623M/ (”Ids < 7. Then a [:0 | f: Mm — y)y'(mo(y))m(y)my(2m.. = /O°° [if—1: +Lw+[;M)fio($ — yl9’("0(y))v(y)dyl2dx 00 M S C j I [MB‘B'x‘”'9’(1(o(y))v(y)dy|2dx+ oo oo 00 -M +2) f (/ (mo(m— ymy)()|dy)2dm+2y2/ (/ (m.(m-y)m(y))dy)2dm <0 f | fmxwy'mnmuwyrdmm /_: (f: [150(133D—v(t/)ldy)2d$ <0 [219:ny / ‘gfly ((yoy)(y)my|2+4y (Imomlmnr’ s c [m e-zfirmmmwHy'(mo())n:.(2M)nmn§ + 472||polliollv||§ _<. Cyny(mo(-))II§.IImui + Maryanne); Thus, this integral converges to 0 uniformly for v E A as a -+ 00, and so T : L2 —+ L2 is compact. Next we show T : H1 —+ H1 is compact. Let 11,, be a bounded sequence in H1. We want to show Tun has a convergent subsequence in H1. Since T is compact in L2, there exists a subsequence, still denoted by un, and u, v E L2 such that Tun —> u, and Tu:l —-> v, as n -—> oo. Define S : L2 -> L2 by Sm a mo . (9”(’U0)ui)ul~ 18 Then S : L2 —+ L2 is compact by the same discussion above because we only need ||g”(u0)|[oo < 00 and g”(uo)uo(:r:) —+ O as a: —> :too. So there exists a subsequence, still denoted by an, and w E L2 such that Sun—>11) inL2, asn—>oo. Therefore for 05 6 08°, n—wo / uqi'dx = lim Tund'dx = — lim (Tun)'qbd:1: = — "lingo/(150 . (y'(mo)m.))'mmm = _[ ”13.30 / (p0 .. (gamma + f(m. . (y"(mo)mz.m.))mmm) = — lim — lim n—boo n—)oo =— = - f(v + w)¢d:c, so u’ = v + w 6 L2, and (Tun)’ —> u’ in L2. Thus, Tun —) u in H1. The claim is proved. CI Lemma 2.2.2 The operator A = I — T is invertible in E. Proof: We only need to show that the equation u — Tu = 0 has no nonzero solutions in E. Note that equation u — Tu = 0 is equivalent to u = 150 * (9'(uo)u). 19 that is —(c2 — d)u" + au = g'(uo)u, (2.10) to which US is a solution. We find the general solution by reduction of order: let u = ugy, then y satisfies III 2ugy' + uoy = 0; let y' = 2, then SO and y = C] + 02/ (UB)—2(S)d8. 1 Thus we get the general solution u = u[)(Cl + C2/12(u[))'2(s)ds). where Cl and 02 are arbitrary constants. Since . f (ma-“vs , lim 113/ (u6)‘2ds = lim 1 = — lim —, 1 :—+00 z—mo 1/u[) x—ioo us which is infinite and we are considering solutions in H‘, C; has to be zero and any H1 solution of (2.10) must be of the form u = Cug, which is odd. Therefore (2.10) has no nonzero solution in E. By the Fredholm alternative, A = I — T is invertible and A‘1 is bounded. D 20 Proof of Theorem 2.2 Fix 0 < 0 < 1 and let Co be positive such that [IA-1|] 5 Co < 00. Since F E C1 6 we can choose 6 > 0 small such that for a constant C1 > 0 satisfying Cl < 2—5, 0 ||DF(u) — DF(uo)|[ 3 Cl, for flu — UOIIHI g 6. Since “DF(U)UHH1 = ”(155 - 150) * (.(I'(U)v)lln1 S Cllpe - polloollvllm, where C depends on [[uIIm, we have IIDFWHI S Cllpm - polls... Note that, because of (A2) (which implies that (3(2) — 1) / 22 is bounded) and flat) — 1 2 | -d€2 - (.(E) I < Id + (mt)2 '6 [6452 +146) + a][(c2 — (1)62 + a] ‘ [(c2 - clot2 + a][(c2 - (1)62 + a1’ [pa - P0] = we have [p5 — p0] -—> O uniformly in E as e —> 0. Thus there exists a positive 50, which depends on 6, such that for any 5 < 80 one has ”DF(U)“ S C2, for ”u — Holly! < 6, 0 .. where Cg < 267' Similar to the discussion in Step 1, F : H1 —+ H 1 maps bounded 0 sets into bounded sets and ”F(mo)” s um. - monsugfifl (I... + I)m'(mo)n..). Hence we can choose small so again, such that ”F(yyom 3 5(1— (9)/C0 for e < 50. By Lemma 2.1.1, u = F (u) + F (u) has a unique solution in the set {u 6 E : [Iu — U0“!!! < 6}. This completes the proof. 21 2.3 Asymptotic behavior of traveling pulses Here we provide an example and show that in this case the traveling pulse decays exponentially at infinity, something that is not obvious for this class of equations and may not be true in general (see [6] section 5). Consider the case j (x) = ge’m. Since 3(6) = 1—+—€—2, then 1 175(5) : 62 i 2 2 _ c 6 1+ 5252 + a and 1 Note that (A2) holds with d 2 d1 = 1. We have the following estimate for the Fourier multipliers: Lemma 2.3.1 Let p1(§) 2 p5 — p0, then 151 decays exponentially and there exist positive constants CE and 'y, where 0.5 -+ 0 as e —+ 0 such that [151] S Gee—Will. Proof: 1 1 pl (6) = 2 — 2 _ 2 0252 _ 6 + a (c 1)§ + a 1 + 6262 c2(§2+a+)(€2+a_) c2 —1 (c2 -1)£2+a ’ where j; 2 _ 2 2 = 771 771 4ac 5 and 771 = c2 — 1 + ae2. 26252 Since c2 > 1 we have that ai > 0. Note that for a > O (e‘alxlsgnx) = __—2ia£ {2 + 012’ 22 and so —i§ 1 1 v _ -¢OiiI| ((52+Oi) 2 ate sgnx. Notice that (1) and = 0(1) when 6 -) 0, and therefore 0+ _ 131 = Cée‘x/‘V'x'sgna: * [e_‘/Elx'sgn$ - (e’fir'z'sgnxl * (V 927-; 6W 32—"..Iml)l ’ where C; —> 0 as e -—> 0. Therefore, there exists '7 > 0 and C5 —> 0 such that I131] S Gee—7'14- Using Lemma 2.3.1 we conclude Theorem 2.3.2 When 3(1):) 2 ée-I’I, the solution us obtained in Theorem 2.1.2 decays exponentially at ioo . Proof: From Lemma 2.3.1, since i). = 131 + 150, v I_’1- pe 0 and '7' = min{7, Mgr-2:1}- Also, since g'(0) = O we have 9(ue(x)) =11m _(__gs) x—yioo 115(13) s—Iino 3 =0. Note that for any x 6 IR Im.(m)|=)(m.my(m.))(m)130;/°° en” “19““ -—“——“y”)( m.(y )Idy —oo “£(y ) and so we draw the conclusion using Lemma 5.3 in [18]: 23 Lemma Suppose that h and k are bounded non—negative functions defined on R with lim k(x) = 0. Let B > 0 and suppose that |x|——>oo m(m) s [_m mfi':-v'k(y)h(y)my for all real x. Then e°"'h E L°° whenever a < 6. Moreover, if K1,K2 and M are constants such that ||h||oo 3 K1, ”It“.>0 3 K2, and k(x) S min{[3/4, (e"° — e‘5)/2)} for [x] 2 M, then Heallh’lloo S 0(aa 182 K19 K21 M)’ 2.4 Instability of traveling pulses In this section we study the instability of those traveling pulses for the nonlocal Klein-Gordon equation discussed earlier. Recall that the equation has the form 1 utt—g(je*u—u)+f(u)=0, for t>0 and xEIR, (2.11) where the kernel j.5 of the convolution is defined by but for srmphcrty, throughout this section we only consuler a particular 3() = -2-e"". 3 We also restrict our attention to the case f (u) = u - u. The traveling pulse (b({) = qb(x — ct) satisfies the second order integro—difi'erential equation eye—E150» m — m) + My) = o. (2.12) 24 Since the traveling pulse ¢(x — ct) is just one mumber of a continuum of traveling pulses obtained by translation, the question of stability or asymptotic stability is broader than it would be for an isolated stationary state. It will be shown that there are solutions with initial data arbitrarily close to (b in L2 which not only leave an L2 neighborhood of ¢(- - ct) for t sufficiently large, but also leave an L2 neighborhood of any fixed translate. That is, there exists 51 > 0, cuch that for all 61 > 0 and k E R, there exists initial data (uo,u1) E L2 x L2 with [[110 —¢||L2 < 61 and |]u1+c¢'||L2 < 61, while |]u(t, x) — ¢(x — ct — k)||L2 > 51 for arbitarily large t > 0, where u(t,x) is the solution to (2.11) with initial values u(0,x) = u0(x), u¢(0,x) = u1(x). (2.13) As we point out in Section 2.1, equation (2.11) can be compared with the classical nonlinear Klein-Gordon equation u“ — u” + f(u) = 0, for t > 0 and x e R, (2.14) and for (2.14) many authors have studied the stability and instability of standing waves, see for example [25, 29, 33]. To show the instability of our traveling pulses for (2.11) we first study the conti- nuity of traveling pulses with respect to the wave speed. To indicate the dependence of P, on c, we use the notation PM. Lemma 2.4.1 uc is continuous with respect to wave speed 0. Proof: Let c1,c2 satisfy cf > Leg > 1, and let um, uc, be the traveling pulses for (2.11) derived from Theorem 2.1.2, with wave speeds c1 and c2, respectively. Then they are fixed points of operators P6,c1 and P5“, respectively. Denote 711 = ”’61 — uO,C1$ a2 = ”62 -' 110,623 25 where 110,6 is the positive, even, homoclinic solution to (2.6), then by (2.7), 111 and 112 are the fixed points of the contraction map Rc(1'1) E (I — DP0,,(u0,,))“(P,,c(a + uofi) — P0,C(uo,c) - DP0,c(uo,C)1‘1), (2.15) for c = c1, c2 respectively, in the set {u E H1(1R): u is even, IIUIIHI < 6}. Then, the contraction property of RC1 implies that ”Um, — “CzllHl = Hue, - “ac, — 1‘62 + ”0,0, + Uo,c1 — Uo,c2“H1 S “”111 — 712HHI + |]u0,c1 — Uo,c2]|H1 = HRqWI) — Reg(fi2)llm + Huacl - ”email”l S ”Remit-11) — Rel (112nml + “Rel (172) — Rc2(fi2)llm +Huo.c1 - u0,czi|Hl S Ollam — mmllym + “Rab-mo — Reg(fi2)llm + Hum... — meallym, where 0 < 1. Therefore, 1 Hum - “czlln1 S 1—_—0 (”Rm (1712) - Izc2(’a2)“H1 + (1+ 6)IIUO,CI _ “0,02HH‘)' Note that uofl = (M x) where 112 is the positive nondegenerate homoclinic l Ve2—1 solution of u” + f (u) = 0, thus 1 1 ||u0,c1 — u0,c2”H1 = [ill/(Wad — W—‘Irliim 2 1 02-1 —+0 asc2—>c1. To Show “1361022) — Rc,(112)]|m —-> 0 as c2 ——> c1, we need the following results: 26 I) |]P5,cl(il2) '— Pe,cz(fi2)”H‘ —) 0 as C2 —> C1. This is true because ”Fatah-‘2) - Pe,cz(fi2)”L2 = ”156m * 1'13 — 156.01 * 1’13]le = ”peg - p6.C2”IF“fi%”L2 —->0 as c2 —> c1, and a similar calculation for the H1 norm. 2) For any v 6 H1) ”D130,“ (umcrlv " DP0,cz(u0.02)v“H’ —) 0 as C2 _> cl- Notice that DP0,C(u0,C)v = 150,, at: (g’(u0,c)v), we can first see in the L2 norm ”DPo,c1(’Uo,c,)v — DPO,C2(u0,C2)v”L2 = “150;, * (9'(Uo,c1)U) - 150.c2 * (9'(Uo,c2)U)IIL2 S ”150“ * (9'(U0,c1)’0) - 130,121 * (£J'(uo,c2)’U)HL2 +“I50,c1 * (g’(u0,C,)v) — 1301:; * (9'(U0,c2)v)“1.2 S “13011 * [(9'(Uo,c.) — g,(u0,cz)lv)”L2 + “(fiOfil — 150,62) * (g,(u0,C2)v)“L2 —)0 as c2 —) c1, and similarly we can show the convergence in H1. 3) Ac E (I — DP0,,,(uo,c))'l is continuous with respect to c. To show this, denote Tc = DPo,c(uo,c). Then (I - TCI)—l — (I "‘ Tczl—l = (I — furl-1“” “ T61+T61" Tar] = (I — Tcil—l — [(1 _ TCI)(I +(1 — TCI)—1(TCI — TC2))l_l 27 and B E —(I — Tc,)“(TCl — Tm) satisfies “B” —+ 0 as c2 —> c1 . Thus for c2 close to Cl) (1 ‘ T.,)-1 -(1 - T02)_1 = (I — Ta)“l — (I — B)‘1(I — n)" = (1"7-21)’l - (1+B+B2+‘”lil-Tc,)’l, : _BU- B)_1(I - Tc1)_l—> 0 3.862 —')Cl. By denOting Sch—1) E P£,c(fl + uO,c) - P0,c(u0,c) " DP0,c(u0,c)l-‘a we get ”Rah—‘2) " RC2 (a2lllH‘ : “AC1361('&2) _ Aczscz(fl2)”ll‘ = “ACISCI(fl'2) - 146130,,(1—12) + 14131522072) " Aczscz (172)”!1‘ S “ACI(SCI — SC2)(fi§-’)”Hl + ”(Am — AC2)562(fi2)“H‘ —~)0 as C; —> c1 because both Operator RC and function no.6 are continuous with respect to c and [ISC,(1“12)||,,1 is bounded as c2 —) c1. Therefore 1 _ _ “um: — Hezllm S l—g-QUIRMW) - Remiuzlllm + (1 + 9)]I'u0m1 - Hamlin!) —-)O as c2 -—> c1, which proves the lemma. Theorem 2.4.2 The traveling pulse of (2.11) derived in Theorem 2.1.2 is unstable. 28 Proof: Let (b = uCl be a traveling pulse corresponding to wave speed c1. Then for l . . 81 E Zl|ucl||L2, and any 61 > 0, by Lemma 2.4.1, there exist a speed c2 Wthh rs close to c, and a traveling pulse ac, for (2.11) such that [IuCl — 1162””! < 61 and ||c1u'cl — cgu'czlle < 61. Without loss of generality, we can assume c2 > c; > 0. Therefore, uc,(x — ct) is a solution of (2.11) and it is initially close to the traveling pulse Uq- However, for any fixed k, there exist M2 > M1 > 0 such that M1 2 1 fl _ ( / may (m — (0) mm) > Tummy: — Jim, and —M2 1 (/ (m..(m - k))’~’dm)m < 5,, thus ||11m(x — clt — k) — u62(x — Cgt)”ig =/ (ucl (x — clt — k) — uc,(x — C2t))2dl' M1+Cxt Z / (uc,(x - clt — k) — uc, (x — c2t))2dx —M1+C1t M1+c1t M1+CI¢ 2 / (ucl (x — clt — k))2dx — / (uc,(x — C2t))2d$ —M1+C1t —Ar11+cjt M1 M1+Clt-Czt 2 f (m... (m — k))2dm — f (m..(m))2mm -A!1 —Afl+C1t—C2t —1\42 2 2m? — / (1., Wm 2 2e? — 5% = 5?, which shows the instability. C] 29 CHAPTER 3 Traveling pulses for the lattice Klein-Gordon equation 3.1 Existence of traveling pulses In this section we study the homoclinic traveling pulse solutions of the lattice system 1 oo 11,, — :2 2 aka.-. + f(un) = 0, n e z (3.1) k=—oo where e > 0 and oz,c satisfies (A3) Zapo, ao<0, ak=a_k, Zakk2=d>0 k=—oo 1:21 and E: [arklk2 = d < 00. 1:31 With the ansatz un(t) = u(en — ct) = u(n), we get the following differential equation with infinitely many advanced and delayed terms: d2u 1 0° 2 , . _ c dnz - 52 aku(n — k8) + f(u) —- O. (3.2) 30 We proceed as before, taking Fourier transforms and reformulating as a fixed point problem. Applying the Fourier transform, equation (3.2) becomes Using (A3) we may write 1 °° . . 1 . . , 25 Z ame'fkfm = 7201(e‘£*‘-2+e‘“"‘)u k=—oo E kZI 4 . 2 ské . E —E§Zak81n(—2—)u. 1:21 Therefore, we may write our equation as ((8 - 2amk28inc2(¥))e + aim = yim‘), 1.21 where sinc(z) = $213. Let 1 (15“.) = 2 k2 . 2 51:6 2 1 (c —:ak srnc (—-2—))§ +a 121 then we can write the equation in the form u : Q€(u)1 where Q; is the operator defined by Q£(u) E q, :1: g(u). Since sinc2(E-’§) = 1 — £5252 + 0(8454), 2 8 q5(£) has the limit 1 130(5) : (C2 _d)€2+a 31 as s —> 0. Therefore, formally when s —+ 0, we have the limit equation 11 = Poi“), where, as before, Po(u) E po :1: g(u); and the integral equation is equivalent to (C:2 - d)U" = —f(u), (3-3) which has the nondegenerate homoclinic solution, uo, discussed previously. By a method similar to that used to study the nonlocal wave equation with a continuous convolution operator, we can again prove the existence of a traveling pulse as near no for c2 > D1, where D1 E sup 2 akkzsinc2(kz) ‘ 1:21 satisfies D1 2 d. Theorem 3.1.1 If conditions (A1) and (A3) are satisfied, then for any speed c satisfying c2 > D1 there exists on so > 0 such that for any s E (0,so) , equation (3.3) has a unique solution us in the set {11 E H1(R) : u is even, ||u — uo||H1 < 6}, where no is the even, positive homoclinic solution of (3.3), and 6 > 0 depends on f, c, and the coefficients 01k, and can be chosen so that 6 < ||uo||H1. Proof: Notice that when 02 > DI, both qE and po are well defined. Therefore, similar to the proof of Theorem 2.2, we only need to show that for a given positive p, there exists a small so > 0 such that llq,S — Polloo < p for s < so. Since Zakkzsingégsg — 1152 mos) — you) = "3‘ 5,, , [(c2 — Z akkzsinc2(—2—))£2 + a][(c2 - d)€2 + a] 1:21 5g and sinc2( 2 ) S 1, there exits {o > 0 such that for [6| > go we have 32 lq - Pol < (D1 + (1)62 E ((C2 - Dl)€2 + a)((C2 - d)? + a) Notice that for |£| S {0, 0 kZI kzi uniformly as s —-> 0 because 2: Iaklk2 < 00. Therefore, q,E —-) po uniformly as s —) 0, 1:21 allowing us to draw the desired conclusion. E] 3.2 Asymptotic behavior of traveling pulses For a lattice with finite range interactions, we give an estimate for the tails of the traveling pulse: Theorem 3.2.1 Suppose that ah = 0 for |k| > ko and suppose c2 > d. There exists 51 > 0 such that for s < 81, the solution u5(z) obtained in Theorem 3.1.1 decays to zero exponentially as 2 —+ :too . Proof: As in Section 2.3, we first show that q} decays exponentially at :l:oo. To estimate the inverse Fourier transform of qs, we need to estimate q5(£) for complex {=x+m. 33 Notice that In + 02le2 = |a + c2(:1:2 — y?) + 2021:31212 = a2 + 2ac2(;1:2 — 312) + 04(232 — y2)2 + 404:1:2y2 = a2 + 2ac2(:2:2 - y2) + 64(132 + 312)2 3 1 202 + ac2x2 _ 3062312 + [Za2 + (162(172 + 312) +c4(:r2 +y2)2] 3 Z Za2 — 3ac2y2 + (g + 62(232 + 312))2 a 2 (5 + c‘*l£l2)2 - E 2_ 2 2 . = a . . if we choose y such that 4[a 3ac y 2 O, i.e., |y| S 7 _ 2lCl. A direct calculation shows that k 1 k |sin2(i-2—§)| = 21-[e’5k3’ + esky — 2] + sin2(E—21—:). Thus, , 2 ska: |sinc2(ik_€)| — —4 ISin2(EE§)l — e—Eky + as” — 2 48m (1.) 2 _ 52k2|€|2 2 - s2k2(:1:2 + y?) 52k2(a:2 + y?)' Since e—sky + esky _ 2 e—sky(esky __ 1)2 52k2y2 = s2k2y2 uniformly for |y| g 7 and lkl 5 ko as s —-> 0, there exists a positive 51 such that for —)1 s 6 (0,51] e‘Eky + 65"” — 2 6*” + eEky -— 2 y2 c2 - d y2 = , < 1 + - . 521:2(132 + 312) 52/33;2 2:2 + y2 ( 2d )12 + y2 Also, it is easily seen that . 5km , skx 4sm2(—2—) _ 4sm2(—2—) $2 < x2 s2k2(:r2 + y'z) _ s2k2$2 $2 + y2 — 2:2 + yz' 34 Therefore we have for lyl g 7, |k| 3 Ito, and s 6 (0,81] ) 3’2 + $2 <1+CZ_J $2+y2 x2+y2— 2d. c2—d 2d k |sinc2(€—2—€-)| S (1+ So, from the definition of qe, for |y| g 7 and s 6 (0,51] l lQEl = — E:akk2sinc2(£fi))é2 + al 2 1:31 1 S ské |c2§2 + a| — IZakkzsinc2(—2—)§2| 1:21 < l _ a c22— d ‘2‘ + C2l€l2 ' Z laklk2(1 +— —d—)l€l2 1:21 _ 1 _ a 2 2 — c d 2 §+cKl-d0+ -)m __ 2 _ a + (62 - (ll—HEP. Thus q€(f) is analytic on the strip Im E < 7. Therefoe, for :1: > 0, level = | / ei"+"”"qe(z)dzl =I/C e‘x‘qe(€+7i)d€| where Q = {z - 7i : -—00 < z < 00}, and by Cauchy’s integral theorem, we have |e'flqel — —|/:e “(gs z + 7i)dz| 0 is small. On the other hand, even though the limit equation does not have homoclinic solutions, since the convolution operator is bounded, we suspect that our models still possess traveling pulse solu- tions with subsonic speed provided 6 is sufficiently large. Our analysis breaks down, however, for 62 < all (or D1 in the lattice case) due to singularities in the transformed kernel p5 (or (15, respectively). 3.3 Instability of traveling pulses In this section we consider the lattice Klein-Gordon equations 1 iin — ;3(un+1- 2a,. + un_1) — un + ui = 0, n E Z (3.4) for which the existence of the traveling pulse was proved in section 3.1. To illustrate the instability we use 4th order Runge-Kutta method solve the Cauchy problem for system (3.4) with the initial data un(0) = ¢o(sn), u;(0) = —c¢o(sn), (3.5) (I! VC2—1 (c2 —1)u”- u+u3 = 0. where ¢o(:r) 2 J2 sech( ) is the solution of the ODE 36 Since we proved that the traveling pulse of (3.4) is in a small neighborhood of ¢o when s is small, we can regard do as a perturbation of the traveling pulse. The numerical simulations show that even though the solution of (3.4) and (3.5) is close to the the traveling pulse initially, it becomes significantly different rapidly. Figures 3.1—3.8 show the solution un(t) of (3.4) at time t = 1,5,6,7,8, 10,15, and t = 20. Figures 3.9-3.16 show the difference between un(t) and ¢o(sn — ct) at time t = 1, 5, 8, 10,15, and t = 20. We use the following choice of parameters in our simulation: N = 1001, —N 1 N—l + , 2 ], s = 0.1, c = 1.1,and At = 0.05. For the boundary conditions, nE[ 2 we choose 2 _ ul+1 “1+2 on the left boundary where l = 43—1 and 2 u —1 um = m um-2 N+1 on the right boundary where m = 2 . 37 Figure 3.1. un(t),t =1 Figure 3.2. un(t),t = 5 38 50 I .0 01 V to 8 Figure 3.3. un(t),t = 6 50 1.5r is 8 Figure 3.4. un(t),t = 7 39 50 1.5" 0.5 » L O Figure 3.5. un(t),t = 8 1.5’ 0.5 - 810 Figure 3.6. un(t),t = 10 40 Figure 3.7. un(t),t = 15 50 1.5r 0.5 '- Figure 3.8. un(t),t = 20 41 Figure 3.9. un(t) — ¢o(sn — ct),t = 1 1.5“ 0.5 * Figure 3.10. un(t) - ¢o(sn — ct),t = 5 42 Figure 3.11. un(t) — ¢o(sn — ct),t = 6 0.5 *- Figure 3.12. un(t) — ¢o(sn — ct),t = 7 43 Figure 3.13. un(t) - ¢o(sn — ct),t = 8 Figure 3.14. un(t) — ¢o(sn — ct),t = 10 44 Figure 3.15. un(t) - ¢o(sn — ct),t = 15 1.5- 0.5 * Figure 3.16. un(t) —- ¢o(sn — ct),t = 20 45 BIBLIOGRAPHY 46 BIBLIOGRAPHY [1] M. J. Ablowitz, Z. H. Musslimani, and G. Biondini, Methods for discrete solitons in nonlinear lattices, Phys. Rev. E, (3) 65 (2002), no. 2, 026602. [2] J. F. R. 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