Using the finite domain remnant of the continuous spectrum to examine integrability: Effect of boundary conditions

Using the finite domain remnant of the continuous spectrum to examine integrability: Effect of boundary conditions Panayotis G. Kevrekidis1and Niurka R. Quintero2 1Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA 2Departamento de Fı ´sica Aplicada I, Escuela Universitaria Polite ´cnica, Universidad de Sevilla, Virgen de A ´frica 7, 41011 Sevilla, Spain and Instituto Carlos I de Fı ´sica Teo ´rica y Computacional, Universidad de Granada, E-18071 Granada, Spain 共Received 11 December 2002; revised manuscript received 30 June 2003; published 23 September 2003兲 The aim of this work is to propose a method for testing the integrability of a model partial differential 共PDE兲 and/or differential difference equation 共DDE兲, by examining it in a finite but large domain. For monoparametric families of PDE/DDE’s, that are known to possess isolated integrable points, we find that very special features occur in the finite domain remnant of the continuous 共‘‘phonon’’兲spectrum at these ‘‘singular’’ points. We identify these features in the case example of a PDE and a DDE 共that sustain front and pulselike solutions, respectively兲for different types of boundary conditions. The key finding of the work is that such spectral features are generic near the singular, integrable points and hence we propose to explore a given PDE/DDE in a finite but large domain for such traits, as a means of assessing its potential integrability. DOI: 10.1103/PhysRevE.68.036612 PACS number共s兲: 05.45.⫺a, 02.30.Ik I. INTRODUCTION Integrable models of partial differential 共PDE兲and differential difference 共DDE兲equations have been a topic of intense investigation over the past few decades 关1–3兴. The main reason for this, except for the wide variety of physical applications that can be described by integrable or nearintegrable systems, is that the special case of integrable models can be analyzed completely by means of the inverse scattering transform 关1,4兴. This can then serve as a starting point for perturbative treatment of near-integrable systems. In the process of these developments, a number of techniques have been developed for assessing integrability in continuous 关5兴or discrete 关6兴settings 共or applicable to both 关7兴兲. An interesting feature of these ‘‘tests’’ is that they are necessary 共but not sufficient兲conditions for integrability. Hence, if a model equation fails such a criterion, it is nonintegrable, but if it passes, it may or may not be integrable. In a sense, this suggests that we still do not understand the essential ingredients that render a system completely integrable. Of course, should a Lax pair be identified and the inverse scattering mechanism be applied, we know that the system is integrable, but it would certainly be desirable 共as is clear from all the above effort to create ‘‘integrability tests’’兲 to have a mechanistic 共‘‘black box’’兲type of criterion to assess that. We, of course, do not claim to be providing a full answer to this question in the present work. However, we will try to give a number of useful hints that may lead to partial answers to the above questions and may provide some intuition in the effort to construct such mechanistic criteria. Our tool of choice will be the use of different sets of boundary conditions 共BC兲to examine the spectrum of the linearization around the nonlinear coherent structure that the PDE/DDE of interest supports. Notice that the effect of boundary conditions in related contexts has been studied in a number of references; see, e.g., Ref. 关8兴, and references therein. However, in all of these works the effects of the BC to the point spectrum were assessed and moreover, this was not done in direct connection with issues of integrability. Here we will, instead, focus on the continuous spectrum; in fact, since we will be dealing with finite but large domains, we will center our attention around the discrete spectrum remnant that ‘‘becomes’’ the continuous spectrum in the infinite domain limit. In the finite domain case, the 共formerly continuous兲spectrum becomes discrete due to the quantization of the wave numbers, imposed by the boundary conditions 共see, e.g., Sec. II below兲. It is exactly this discrete remnant of the continuous spectrum, that we aim at examining here, to elucidate its interesting properties in integrable versus nonintegrable settings. In the present work, we focus on two model problems, to establish our findings and demonstrate their generality. The models are selected as one-parameter families of equations such that one member of the family is an integrable system. Moreover, in illustrating the generality of the conclusions, they are selected in a form such that one model corresponds to a PDE, while the other to a DDE, so that one is kink bearing, while the other is pulse bearing. The models of interest will be the parametrically modified sine-Gordon equation 关often also called the Peyrard-Remoissenet 共PR兲model兴 关9,10兴and a modified version of the discrete nonlinear Schro ¨dinger 共DNLS兲model 共occasionally called the Salerno model兲关11兴. The former PDE reads ␾ tt⫺ ␾ xx⫽⫺ dU d ␾ ,U共 ␾ ,r兲⫽共1⫺r兲2关1⫺cos共 ␾ 兲兴 1⫹r2⫹2rcos共 ␾ 兲 共1兲 in the infinite domain 兩 x 兩 ⬍⬁and with 兩 r 兩 ⬍1; while the latter DDE is of the form iu ˙n⫽⫺⌬2un⫺ 兩 un 兩 2关2 ⑀ un⫹共1⫺ ⑀ 兲共un⫹1⫹un⫺1兲兴.共2兲 The most well known among these monoparametric families of models are the sine-Gordon equation 关Eq. 共1兲, for r⫽0] which is relevant to superconductivity and charge density PHYSICAL REVIEW E 68, 036612 共2003兲 1063-651X/2003/68共3兲/036612共7兲/$20.00 ©2003 The American Physical Society68 036612-1 waves among other applications 关2兴and the experimentally realizable discrete nonlinear Schro ¨dinger equation 关12兴of ⑀ ⫽1, as well as its integrable, so-called Ablowitz-Ladik 关13兴 counterpart for ⑀ ⫽0 in the case of Eq. 共2兲. Notice that for the PDE, the subscripts denote partial derivatives of the field, while for the DDE, the overdot denotes temporal derivative, ⌬2un⬅C(un⫹1⫺2un⫹un⫺1), where C⫽1/(⌬x)2is a constant determined by the lattice spacing ⌬x; the subscript ndenotes the lattice site index. In the former case, there exist kinklike solutions which have been detailed in Refs. 关9,10兴, while in the latter, the field is complex and there exist pulselike solutions of the form un ⫽exp(i⌳t)vn, where ⌳is the frequency of the solutions and vnits 共real兲exponentially localized spatial profile 关11,12兴. In the PDE, linearization around a state ␾ 0(x), using the ansatz ␾ ⫽ ␾ 0(x)⫹ ␦ exp(i ␻ t)f(x) into Eq. 共1兲, yields to O( ␦ ) the linearization equation fxx⫹关 ␻ 2⫺U⬙共 ␾ 0,r兲兴f⫽0. 共3兲 Notice that when r⫽0共in the infinite domain limit兲, ␾ 0(x) ⫽4 arctan关exp(x)兴is the static kink solution of the sG equation and for this function, the Sturm-Liouville problem 共3兲 can be exactly solved 关14兴yielding one discrete mode 共Goldstone mode兲at ␻ ⫽0 and the continuous spectrum represented by the phonons, ␻ k⫽ 冑 1⫹k2,fk共x兲⫽exp共ikx兲 冑 2 ␲␻ k 关k⫹itanh共x兲兴,共4兲 for all values of k. For r⫽0, neither the static solution nor the linearization spectrum are explicitly available in the infinite domain limit. Analogously to the PDE, for the linear stability analysis of DDE 共2兲we insert exp(i⌳t)关vn⫹ ␦ (Une⫺i ␻ t⫹Wnei ␻ 쐓t)兴into Eq. 共2兲. We thus obtain to O( ␦ ) the following eigenvalue problem for 兵 ␻ , 兵 Un,Wn 쐓 其 其: ␻ 冉 Un Wn 쐓 冊 ⫽L 冉 Un Wn 쐓 冊 ,L⫽ 冉 AB ⫺B⫺A 冊 , Amn⫽关⌳⫹2C⫺ 兵 4 ⑀ vn 2⫹共1⫺ ⑀ 兲vn关vn⫹1⫹vn⫺1兴 其 兴 ␦ m,n ⫹关共1⫺ ⑀ 兲vn 2⫺C兴共 ␦ m,n⫹1⫹ ␦ m,n⫺1兲, Bmn⫽⫺vn关2 ⑀ vn⫹共1⫺ ⑀ 兲共vn⫹1⫹vn⫺1兲兴 ␦ m,n,共5兲 where the stars denote complex conjugation. The paper is organized as follow. In the following section we obtain an approximate solution for the Sturm-Liouville problem 共3兲by imposing different types of boundary conditions in the finite domain of length L. The obtained results are compared with the numerical computations in Sec. III, where we also computed the solution of Eq. 共5兲. Finally, we summarize our findings and present our conclusions in Sec. IV. II. ANALYTICAL APPROXIMATION In this section we solve approximately Eq. 共3兲when 兩 x 兩 ⬍L/2, where Lis the finite 共but large enough兲length of the system. Notice that our results will be generically true, if Lis chosen large enough. By large enough here, we mean a domain size which is many times 共at least 10兲larger than the characteristic length of the solitary wave 共kink or pulse兲that we will examine inside this domain. We will take into account different kinds of BC, in particular, free fx共⫺L/2兲⫽0, fx共L/2兲⫽0, 共6兲 fixed f共⫺L/2兲⫽0, f共L/2兲⫽0, 共7兲 and antiperiodic boundary condition 共aPBC兲 fx共⫺L/2兲⫽⫺fx共L/2兲,f共⫺L/2兲⫽⫺f共L/2兲.共8兲 First we consider the integrable case, r⫽0, and we show that for the first phonon modes, the eigenfrequencies ␻ ˜ n free ⫽ ␻ ˜ n⫺1 fixed and ␻ ˜ n ap have a double multiplicity 共we will denote with tilde the analytical, approximated eigenfrequencies兲.To proceed, we use the exact solution of problem 共3兲for r⫽0in the infinite domain. We would like to stress that if we change the infinite domain by a finite one, with a given BC, we will still have an infinite number of eigenfrequencies,1but for the allowed wave numbers k关15兴. In order to calculate approximately these allowed wave numbers, we proceed as in Ref. 关16兴. Notice that fk(x)⫽Fk(x)⫹iGk(x), where Fk共x兲⫽kcos共kx兲⫺sin共kx兲tanh共x兲 冑 2 ␲␻ k ,共9兲 Gk共x兲⫽cos共kx兲tanh共x兲⫹ksin共kx兲 冑 2 ␲␻ k .共10兲 Then the solution of Eq. 共3兲, with r⫽0, related to the phonon contribution is represented by the linear superposition of all the odd 关Gk(x)兴and even 关Fk(x)兴phonon modes ␺ 共x,t兲⫽兺 k关ak共t兲Fk共x兲⫹bk共t兲Gk共x兲兴.共11兲 Imposing free BC for each phonon mode of Eq. 共11兲we obtain that the first wave numbers satisfy ak共t兲关sin共kL/2兲关k2⫹cosh⫺2共L/2兲兴⫹kcos共kL/2兲tanh共L/2兲兴 ⫽0, 共12兲 bk共t兲关cos共kL/2兲关k2⫹cosh⫺2共L/2兲兴⫺ksin共kL/2兲tanh共L/2兲兴 ⫽0. 共13兲 1Note that this is true for the continuum problem of Eq. 共3兲, but would no longer be true for the discrete one of Eq. 共5兲. P. G. KEVREKIDIS AND N. R. QUINTERO PHYSICAL REVIEW E 68, 036612 共2003兲 036612-2 The solutions of these transcendental equations yield the allowed values of k. We can solve these approximately if we consider LⰇ1. Then, we find that kn,0 free⫽n⫺1 L ␲ ,n⫽1,2,3,..., nⰆL,共14兲 where the zero subscript denotes that we are dealing with the unperturbed case r⫽0, and its corresponding eigenfunctions are related with the odd functions Gn(x) for the odd numbers nand with the even functions Fn(x) for the even numbers n. Hence, the first eigenfrequencies are represented by ␻ ˜ n,0 free⫽ 冑 1⫹ 冉 n⫺1 L ␲ 冊 2 ,n⫽1,2,..., nⰆL. 共15兲 Analogously, for fixed BC the following relations hold; ak共t兲关kcos共kL/2兲⫺sin共kL/2兲tanh共L/2兲兴⫽0, 共16兲 bk共t兲关cos共kL/2兲tanh共L/2兲⫹ksin共kL/2兲兴⫽0. 共17兲 Then, for large enough L, we find that kn,0 fixed⫽n L ␲ ,n⫽1,2,3,..., nⰆL,共18兲 and so, ␻ ˜ n,0 fixed⫽ 冑 1⫹ 冉 n L ␲ 冊 2 ,n⫽1,2,..., nⰆL,共19兲 where the odd 共even兲numbers nare related with the odd Gn(x)关even Fn(x)] eigenfunctions. Remark 1. By comparing expressions 共15兲and 共19兲we observe that in the integrable case (r⫽0) ␻ ˜ n free⫽ ␻ ˜ n⫺1 fixed for the first few eigenfrequencies. Now by imposing aPBC in each phonon mode of Eq. 共11兲 and taking into account the symmetry properties of Fk(x), Gk(x) and their derivatives, the equations that the wave number satisfies can be reduced to akFk共L/2兲⫽0, 共20兲 bk ⳵ Gk ⳵ x共L/2兲⫽0. 共21兲 Notice that Eqs. 共20兲and 共21兲coincide with Eqs. 共16兲共i.e., the first equation for fixed BC兲and 共13兲共i.e., the second equation for free BC兲, respectively. The solutions of Eqs. 共20兲and 共21兲are given by kn,0 ap⫽2共n⫺1兲 ␲ L,n⫽2,3,..., nⰆL,共22兲 kn,0 ap⫽2共n⫺1兲 ␲ L,n⫽1,2,3,..., nⰆL,共23兲 respectively, and their eigenfunctions correspond to the even Fn(x) and odd Gn(x). Then, the first eigenfrequencies are represented by ␻ ˜ n,0 ap⫽ 冑 1⫹ 冉 2共n⫺1兲 ␲ L 冊 2 ,n⫽2,3..., nⰆL,共24兲 ␻ ˜ n,0 ap⫽ 冑 1⫹ 冉 2共n⫺1兲 ␲ L 冊 2 ,n⫽1,2,..., nⰆL. 共25兲 This means that the even 共odd兲modes for aPBC 兵 ␻ ˜ n ap ,an(t)Fn(x) 其 关 兵 ␻ ˜ n ap ,bn(t)Gn(x) 其 兴coincide with the even modes for fixed BC 共odd modes for free BC兲. Remark 2. From relations 共24兲and 共25兲we conclude that for the integrable case and aPBC the eigenfrequencies have multiplicity 2. The analysis of the Sturm-Liouville problem 共3兲for the nonintegrable case, r⫽0, becomes more complicated since ␾ 0(x) is the exact kink solution of Eq. 共1兲and this function is only known in the implicit form 关9兴共even for the infinite domain problem兲. So, instead of solving this equation we calculate approximately the solution of 冋 d2 dx2⫺V共x兲⫺rW共x兲⫹E 册 f⫽0, 共26兲 where V(x)⫽⫺2/cosh2(x), W(x)⫽8 tanh(x)关x ⫺5tanh(x)兴/cosh2(x), and E⫽ ␻ 2⫺ ␻ ph 2with ␻ ph⫽(1 ⫺r)/(1⫹r)关10,17兴. This eigenvalue problem is obtained in two steps: first we find a solution for small rof Eq. 共1兲, through the perturbative expansion ␾ (x,t)⫽ ␾ sG(x) ⫹r ␾ 1(x)⫹O(r2), where ␾ sG(x) is the static sG kink and second we linearize Eq. 共1兲around the obtained solution up to order of r, so we insert ␾ (x,t)⫽ ␾ sG(x)⫹r ␾ 1(x) ⫹ ␦ 关f(x)exp(i ␻ t)⫹f쐓(x)exp(⫺i ␻ t)兴into Eq. 共1兲and consider the equation that arises to O( ␦ ) and obtain Eq. 共26兲. Arguably, this approach fails to capture the corrections to the tail of the wave due to domain finiteness. However, as argued in Ref. 关8兴, the latter are exponentially small in the length of the domain. Hence, as will also be justified a posteriori, here we capture the leading order dependence in L, as well as the leading order effect of r关see, e.g., Eqs. 共31兲–共33兲below兴. Then, following the procedure of the perturbation methods for linear eigenvalue problem suggested in Ref. 关18兴,weassume the solution of Eq. 共26兲as En⫽En,0⫹rEn,1⫹O共r2兲,共27兲 fn共x兲⫽fn,0共x兲⫹rfn,1共x兲⫹O共r2兲,共28兲 where the first subscript in the functions, n, denotes the order of the phonon modes 共for r⬍0 this subscript can also denote the internal mode兲, the second one corresponds to the order of perturbation. By inserting these expansions in Eq. 共26兲 and equating and collecting the terms of the same order in r, we obtain for O(r0), USING THE FINITE DOMAIN REMNANT OF THE... PHYSICAL REVIEW E 68, 036612 共2003兲 036612-3 冋 d2 dx2⫺V共x兲⫹En,0 册 fn,0⫽0, 共29兲 and for the next order correction O(r1), 冋 d2 dx2⫺V共x兲⫹En,0 册 fn,1⫽关W共x兲⫺En,1兴fn,0 .共30兲 Notice that Eq. 共29兲corresponds to the integrable case r ⫽0 already solved for free 关Eq. 共15兲兴, fixed 关Eq. 共19兲兴, and antiperiodic BC 关see Eqs. 共24兲and 共25兲兴. Notice also that En,0⫽ ␻ ˜ n,0 2⫺1⫽kn,0 2and that its corresponding eigenfunction fn,0(x) is related either with the odd Gn(x) or even Fn(x). Then, for different boundary conditions, the eigenfrequencies of Eq. 共26兲are determined by ␻ ˜ n free⫽ 冑 ␻ ph 2⫹共kn,0 free兲2⫹rEn,1 free,共31兲 ␻ ˜ n fixed⫽ 冑 ␻ ph 2⫹共kn,0 fixed兲2⫹rEn,1 fixed,共32兲 ␻ ˜ n ap⫽ 冑 ␻ ph 2⫹共kn,0 ap兲2⫹rEn,1 ap,共33兲 where kn,0 free ,kn,0 fixed , and kn,0 ap are given by Eqs. 共14兲,共18兲, and 共22兲and 共23兲, respectively. The solution of the eigenvalue En,1 for the first-order correction is given by En,1⫽ 冕 ⫺L/2 L/2 dxfn,0共x兲W共x兲fn,0共x兲 冕 ⫺L/2 L/2 dxfn,0 2共x兲 .共34兲 The integrals involved in Eq. 共34兲can be computed numerically for different BC and different values of r( 兩 r 兩 Ⰶ1), then we can calculate the approximated eigenfrequencies in each case. We can now compare these results with the numerical solutions of Eq. 共3兲关for details on the numerical methods/ results, we refer the reader to Sec. III兴. From the data of the Tables I and II we observe an oscillatory behavior of ␻ ˜ n free⫺ ␻ ˜ n⫺1 fixed for the first phonon’s modes for r⫽0. We also notice that the eigenvalues for aPBC lose their double multiplicity that existed in the case of the integrable equation. It is also worth noting that these features are typically observable in the third decimal digit of the corresponding eigenfrequencies. On the other hand, the difference 共well justified within the approximations mentioned above兲between the theoretical and numerical predictions for the individual eigenfrequencies is typically in the fourth or fifth decimal digit. Hence, the observations of the previous paragraph are systematic and in agreement with the theoretical predictions. III. NUMERICAL RESULTS AND DISCUSSIONS To find the numerical solution of Eqs. 共1兲and 共3兲,we discretize the equations in a numerical mesh for a finite domain. The mesh consists of the N⫹1 points xj⫽ 兵 ⫺L/2 ⫹j⌬x,j⫽0,1,2,...,N 其 defined in the finite length Lof the system (⌬x⫽L/N). Notice that since, in this case, we wish to emulate the behavior of the PDE, ⌬xis very fine 共typically 0.05), and the robustness of the findings upon variation of the 共small兲⌬xhas been verified. When we compute the solution either of the PDE or of the linearization equation, we consider three different types of BC 共6兲–共8兲. We would like to remark that this kind of discretization of the SturmLiouville problem 共3兲only affects the last phonon modes, so we can compare the behavior of the first phonon modes obtained in the preceding section with the numerical solution of Eq. 共3兲. In both cases, the distributions of these eigenfrequencies are determined by the parameter rand by the different boundary conditions in the finite domain. TABLE I. For positive and small value of r⫽0.02, we compare the first eigenfrequencies, obtained perturbatively, ␻ ˜ n, with the ones computed by solving the original Eqs. 共1兲and 共3兲, ␻ n. n ␻ n free ␻ ˜ n free ␻ n fixed ␻ ˜ n fixed ␻ n ap ␻ ˜ n ap 1 0.96117 0.96192 0.96234 0.96352 0.96117 0.96192 2 0.96132 0.96129 0.96293 0.96284 0.96293 0.96284 3 0.96436 0.96514 0.96729 0.96760 0.96436 0.96514 4 0.96560 0.96541 0.96933 0.96899 0.96933 0.96899 5 0.97118 0.97126 0.97604 0.97573 0.97118 0.97126 6 0.97411 0.97358 0.97989 0.97915 0.97991 0.97915 TABLE II. We provide the same comparison as in the previous table for a negative value of r⫽ ⫺0.02. Here, ⍀ ˜ iand ⍀irepresent the internal mode calculated by the perturbation method and computed by numerical solution of Eq. 共3兲, respectively. n ␻ n free ␻ ˜ n free ␻ n fixed ␻ ˜ n fixed ␻ n ap ␻ ˜ n ap ⍀i⫽1.03560 ⍀ ˜ i⫽1.03977 ⍀i⫽1.03560 ⍀ ˜ i⫽1.03924 ⍀i⫽1.03560 ⍀ ˜ i⫽1.03977 1 1.04131 1.04129 1.04278 1.04271 1.04156 1.04058 2 1.04156 1.04058 1.04367 1.04303 1.04278 1.04271 3 1.04524 1.04506 1.04866 1.04835 1.04694 1.04625 4 1.04694 1.04625 1.05125 1.05057 1.04867 1.04835 5 1.05306 1.05257 1.05838 1.05770 1.05657 1.05564 P. G. KEVREKIDIS AND N. R. QUINTERO PHYSICAL REVIEW E 68, 036612 共2003兲 036612-4 We also compute the solutions of DDE 共2兲and Eq. 共5兲 using 200 points and ⌬x⫽0.75. The BC are defined analogously through U0⫽U1,UN⫽UN⫺1,W0⫽W1, and WN ⫽WN⫺1for free BC. For fixed BC: U0⫽0, UN⫽0, W0 ⫽0, and WN⫽0, while for periodic BC: U0⫽UN⫺1,UN ⫽U1,W0⫽WN⫺1, and WN⫽W1. Our results when the parameter of the PR potential or ⑀ in the DDE are varied can be summarized in Figs. 1–6. From the above results, the following conclusions can be drawn. 共1兲For fixed BC, the band edge frequency is prohibited. Hence, we compare ␻ n free with ␻ n⫺1 fixed . We find that for small wave numbers, fixed and free BC eigenfrequencies practically coincide only in the integrable case, whereas for the nonintegrable case we observe an oscillatory behavior of this function 关see Figs. 1 and 2兴. In Fig. 2 we also show the oscillatory behavior of ␻ ˜ n free⫺ ␻ ˜ n⫺1 fixed , obtained from the perturbation theory, for r⫽⫺0.02 共open triangles兲and r⫽0.02 共open squares兲. 共2兲For antiperiodic BC, the spectrum comprises of modes coming alternately from the free and fixed BC. This seems natural as the free boundary conditions select eigenmodes FIG. 1. Comparison of the eigenfrequencies of the 共discrete remnant of the兲continuous spectrum for fixed and free BC: We have plotted the difference between the eigenfrequencies computed from Eq. 共3兲and ␻ ph⫽(1⫺r)/(1⫹r)vsr. The circles joined by solid line 共free BC兲represent how far the frequencies are from the lower phonon mode. The triangles joined by dotted lines correspond to fixed BC. FIG. 2. The difference between the first frequencies for free and fixed BC, ␻ n free⫺ ␻ n⫺1 fixed (2⭐n⭐20), is plotted as a function of the wave number for r⫽0共circles joined by solid line兲,r⫽⫺0.02 共triangles joined by dashed line兲, and r⫽0.02 共squares joined by dotted line兲. The open triangles (r⫽⫺0.02) and squares (r⫽0.02) represent the differences between the frequencies obtained by the perturbation theory, ␻ ˜ n free⫺ ␻ ˜ n⫺1 fixed , in the preceding section. FIG. 3. Antiperiodic BC: the difference between the numerical eigenfrequencies computed from Eq. 共3兲and the band edge of the 共formerly continuous兲spectrum ␻ ph⫽(1⫺r)/(1⫹r) is shown. Adjacent eigenmodes are given by circles joined by solid line and triangles joined by dotted line. The relevant internal mode is shown by circles joined by solid line 共the first curve from below兲. FIG. 4. Antiperiodic BC: We show the difference between ␻ n period⫺ ␻ n⫹1 period vs n(n⫽2,4,...,20). The stars practically at zero for all nrepresent the integrable system (r⫽0), whereas the long-dashed (r⫽⫺0.02) and dotted (r⫽0.02) lines correspond to nonintegrable cases 共numerical results兲. With triangles (r⫽ ⫺0.02) and circles (r⫽0.02) we plot the eigenfrequencies obtained from the perturbation theory 共analytical results兲. USING THE FINITE DOMAIN REMNANT OF THE... PHYSICAL REVIEW E 68, 036612 共2003兲 036612-5 symmetric at the boundary, the fixed ones select modes antisymmetric at the boundary, while the antiperiodic BC allow for both 共cf. Figs. 3 and 4兲. 共3兲An additional feature, equally important as 共1兲共especially in view of its potential predictive power兲is the fact that for the integrable case of r⫽0, antiperiodic BC essentially imply the presence of double eigenvalues. The difference between the two eigenvalues is O(10⫺9) for all pairs 共except for the cutoff, discretization induced phenomena at the upper end of the spectrum which are irrelevant兲. This is in sharp contrast 共in particular, for small wave numbers兲,to even mild breakings of integrability, as can be inferred from Fig. 4. 共4兲Statements 共1兲and 共3兲above can be used in predictive form and constitute the criterion 共algorithm兲set forth in this work: for a given PDE/DDE model, we find the steady state coherent structure 共i.e., solitary wave兲in a finite but large domain. This can be done, e.g., by finding the exact solution of an ODE or numerically performing a Newton-type algorithm. Linearize around the exact, finite domain solution and study, in particular, the small wave numbers, close to the lower edge of the spectrum 共we assume that the problem is monoparametric in what follows, but it is clear that the application of the criterion does not require that兲. If for a critical/singular value of the parameter the fixed BC and free BC 共small k) eigenvalue spectra 共of the remnant of what for the infinite domain was the continuous spectrum兲essentially coincide and the multiplicity of antiperiodic BC eigenvalues becomes double, then the model for this unique value of the parameter can be ‘‘strongly suspected’’ to be integrable. We use the above expression, as we provide no rigorous proof, but only supporting 共but rather universal in distinct models with distinct features/solutions兲numerical evidence for this statement. 共5兲We have also tested the validity of these results in Eq. 共2兲, in the vicinity of the integrable limit ⑀ ⫽0, with similar conclusions 关see Figs. 5 and 6兴. Indeed, in Fig. 5 we observe the oscillatory behavior of ␻ n free⫺ ␻ n⫺1 fixed in the nonintegrable case, in Fig. 6, we show the case of periodic BC, where it can be clearly seen that it is only for the integrable case that the double eigenvalue multiplicity is obtained. IV. CONCLUSIONS In conclusion, we have proposed and used a test for revealing the potential integrable nature of a given model problem. By varying the boundary conditions of a finite domain computation and examining the effects of such variations in the 共continuous-turned-discrete兲spectrum, we have revealed that the small wave numbers have singular ways of responding to the unique parameter values for which the model is integrable. These singular features 关such as an approximate identification of fixed with free BC for small keigenvalues and the double multiplicity of eigenvalues for periodic 共or antiperiodic兲BC兴can be used to identify and single out the integrable behavior. We have provided two model examples, respectively, for kinks and pulses and for a PDE and a DDE. Independently of the detailed structure of the model these properties have been identified as universal and have been supported also by analytical considerations. It would naturally be of interest to explore the potential usefulness of such a criterion in various more complex settings. ACKNOWLEDGMENTS We would like to thank Jesu ´sSa ´nchez-Dehesa for the useful discussion on the perturbation theory in the SturmLiouville problem. This work has been supported by the Ministerio de Ciencia y Tecnologı ´a of Spain through Grant No. BFM2001-3878-C02 and by the Junta de Andalucı ´a under Project No. FQM-0207 共N.R.Q.兲. It has also been partially supported by NSF under Grant No. DMS-0204585, a University of Massachusetts Faculty Research Grant and the Eppley Foundation for Research 共P.G.K.兲. FIG. 5. The oscillatory behavior of ␻ n free⫺ ␻ n⫺1 fixed in the nonintegrable case ( ⑀ ⫽0.1) is shown for the first wave numbers 共see the squares joined by dashed line兲. The circles joined by solid line are the results for the integrable system ( ⑀ ⫽0). FIG. 6. Periodic BC for AL-DNLS 共Ablowitz-Ladik DNLS兲of Eq. 共2兲: The solid line at zero represents the difference between two consecutive frequencies (n⫽2,4,...)fortheintegrable AL lattice ( ⑀ ⫽0). The double multiplicity of the frequencies is destroyed as ⑀ is increased 共dotted, dashed, and dot-dashed lines represent the nonintegrable cases of ⑀ ⫽0.1,0.5,1, respectively兲. P. G. KEVREKIDIS AND N. R. QUINTERO PHYSICAL REVIEW E 68, 036612 共2003兲 036612-6 关1兴M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform 共SIAM, Philadelphia, 1981兲. 关2兴R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, and H.C. Morris, Solitons and Nonlinear Wave Equations 共Academic Press, London, 1982兲. 关3兴A.C. Scott, Nonlinear Science 共Oxford University Press, Oxford, 1999兲. 关4兴L.D. Faddeev and L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons 共Springer-Verlag, Berlin, 1987兲. 关5兴P. Painleve ´, C. R. Acad. Sci. 共Paris兲130, 1112 共1900兲; A. Ramani, B. Grammaticos, and A. Bountis, Phys. 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