CMSV-264(online)-Panayotaros.pdf

Transcription

1 COMMUN. MATH. SCI. Vol. 15, No. 8, pp c 017 International Press PROPERTIES OF SOME BREATHER SOLUTIONS OF A NONLOCAL DISCRETE NLS EQUATION ROBERTO I. BEN, JUAN PABLO BORGNA, AND PANAYOTIS PANAYOTAROS Abstract. We present results on breather solutions of a discrete nonlinear Schrödinger equation with a cubic Hartree-type nonlinearity that models laser light propagation in waveguide arrays that use a nematic liquid crystal substratum. A recent study of that model by Ben et al [R.I. Ben, L. Cisneros Ake, A.A. Minzoni, and P. Panayotaros, Phys. Lett. A, 379:1705C-1714, 015] showed that nonlocality leads to some novel properties such as the existence of orbitaly stable breathers with internal modes, and of shelf-like configurations with maxima at the interface. In this work we present rigorous results on these phenomena and consider some more general solutions. First, we study energy minimizing breathers, showing existence as well as symmetry and monotonicity properties. We also prove results on the spectrum of the linearization around one-peak breathers, solutions that are expected to coincide with minimizers in the regime of small linear intersite coupling. A second set of results concerns shelftype breather solutions that may be thought of as limits of solutions examined in [R.I. Ben, L. Cisneros Ake, A.A. Minzoni, and P. Panayotaros, Phys. Lett. A, 379:1705C-1714, 015]. We show the existence of solutions with a non-monotonic front-like shape and justify computations of the essential spectrum of the linearization around these solutions in the local and nonlocal cases. Keywords. discrete nonlinear Schrödinger equations; nonlocal effects; localized solutions; breather solutions; linear stability. AMS subject classifications. 37K60; 35Q55; 47J10; 47J30; 78A Introduction We present results on breather solutions of a nonlocal discrete nonlinear Schrödinger equation with a cubic Hartree-type nonlinearity introduced by Fratalocchi and Assanto [7] to study the propagation of laser light in waveguide arrays built from a nematic liquid crystal substratum. Further experimental work on this system is reported in [8, 10, 9]. Breather solutions of discrete NLS equations have both theoretical and experimental importance and have been studied extensively, especially for the discrete NLS equation with power nonlinearities, see [16] and references. Recently Ben et al [1] reported some interesting new properties of breather solutions of the nonlocal discrete NLS system of [7] in a 1-D lattice, specifically the presence of internal modes around orbitaly stable breathers, and the appearance of front-like solutions that attain their maximum at the interface. The present work studies some of these properties from a rigorous point of view, and includes new related results on breathers of both the local and nonlocal discrete cubic NLS equations. We will show existence and stability results for two types of solutions of the Fratalocchi-Assanto [7] system. First, we study breather solutions that minimize the Hamiltonian (energy) over configurations with fixed l norm (power). These are spatially localized solutions that could be in principle studied experimentally, see [8,10,9] for some related localized solutions. The existence of the energy minimizers follows from a concentration-compactness argument that is similar to the one used in [0,,30,3]. Received: May 3, 016; accepted (in revised form): July 3, 017. Communicated by David Cai. Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento, J.M. Gutiérrez 1150, 1613 Los Polvorines, Argentina (benroberto@gmail.com). CONICET, Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M. Gutierrez 1150, Los Polvorines, Provincia de Buenos Aires, Argentina (jpborg@gmail.com). Depto. Matemáticas y Mecánica, I.I.M.A.S., Universidad Nacional Autónoma de México, Apdo. Postal 0-16, Cd. México, México (panos@mym.iimas.unam.mx). 143

2 144 BREATHERS OF NONLOCAL DISCRETE NLS We also show that the minimizer is even and decays monotonically away from its maximum. These two properties follow from discrete symmetrization and rearrangement arguments, see Theorem. for statements and proofs. The notion of symmetrization for functions on the lattice uses the Fourier transform, while the monotonicity argument uses a discrete analogue of Riesz s rearrangement inequality, see [17]. We note that [1] shows the existence of spatially decaying breathers using different arguments, assuming that the linear intersite coupling parameter δ is small in absolute value. The present results assume instead that δγ >0, see (.1),andthusextendthe existence results of [1] to a larger parameter range. The δγ >0 case is also the one relevant to experiments. The numerical results of [1] suggest that the linearization around the minimizing solutions of section should have internal modes. Showing this in the general case seems difficult and we here present results for the linearization around one-peak breather solutions obtained for small linear intersite coupling δ via a continuation argument, see [1]. These solutions exist in the intersection of the parameter ranges considered in Sectionandin[1], and we argue that they coincide with the energy minimizers of Section, see Proposition 4.3 for the precise statement. The presence of stable internal modes was seen explicitly in the limiting case on vanishing linear coupling δ in [1]. In Section 4 we use perturbation arguments, especially the Krein signature, to show the persistence of at least a finite number of stable internal modes for δ>0 sufficiently small, see Proposition 4.. These results agree with the numerical observations of [1]. The second type of solutions we study are shelf-type breather solutions. These are solutions that decay in one direction and asymptote to a nonzero value in the other direction. These solutions have infinite l -norm, and their existence is shown by an implicit function theorem that allows us to continue solutions of the δ = 0 equation, see [1, 18, 4, 7] for related results for solutions that decay at infinity. The argument applies to both the power and nonlocal nonlinearities, see Section 3. The power case is here primarily interesting as a limit of the nonlocal case. In the nonlocal case we continue from a nontrivial configuration that exhibits a maximum at the interface between sites with vanishing and nonvanishing amplitude. This effect was seen originally in [1] forl solutions, and we see here that the linearization can be still inverted explicitly. We also consider the linearization around the shelf-type breathers of Section 3 and present results on the essential spectrum of the corresponding operators. In the limiting case of the power nonlinearity, the essential spectrum is obtained by studying the spatial dynamics of the piecewise constant coefficient linear map defined by the linear operator. This is a common approach for problems on the line, see e.g. [14], and we show discrete analogues of some of the basic results in Subsection 4.. In the nonlocal case we do not seem to have an immediate correspondence of the operator problem to spatial dynamics, but we can reproduce some of the calculations of the local case using the fact that the operator problem involves piecewise translation invariant (convolution) operators to obtain a weaker result for the essential spectrum, see Theorem 4. in Subsection 4.3. A possible alternative approach may be provided by a study of the analytic properties of the resolvent, see [1] for discrete Schrödinger operators, and [8] for the linearization around NLS breathers. We note that the motivation for studying the shelf-type solutions is heuristic at present. First, the shelf-type breathers can be thought of as limits of a class of breathers that have finite support in the δ = 0 limit. see [1]. These finite shelf-type breather solutions can be characterized by an integer m>0, the size of their support in the δ = 0 limit. The essential spectrum computed in Section 4 is the apparent limit of a

3 ROBERTO I. BEN, JUAN PABLO BORGNA, AND PANAYOTIS PANAYOTAROS 145 subset of the spectrum of the linearization around the finite shelf-type breathers as m increases, see details in Section 4. We also point out that there are interesting connections between shelf-type breathers of discrete NLS equations and static front solutions of discrete nonlinear diffusion equations. The equations for these solutions are similar, and the continuation methods of Section 3 can be also used to show existence and monotonicity properties of static fronts, see [4, 11]. In Section 3 we remark on the discrete version of a recently proposed diffusion equation with nonlocal nonlinearity, see [9]. Also, in discrete diffusion equations the static solutions can become depinned, i.e. may start to move after some value of the intersite coupling, and the static problem is a starting point for analyzing these moving fronts []. In the discrete NLS we do not have a clear analogue of the depinning transition for shelf-type solutions, but a preliminary numerical study [1] suggests that such a transition may be possible. This transition can be related to changes in the spectrum and bifurcations of the shelf-type breathers as we increase the linear coupling. The Fratalocchi Assanto model can be generalized to higher dimensions, see e.g. [31, 3] for dimension dependent phenomena in the local system, however the experimental setup of [7, 8, 10, 9] does not have a higher dimensional analogue. Also, comparisons of the model to experiments have used δ, γ, κ near unity, see [7], and thus the results of Section are directly relevant. The small δ results of Sections 3, 4 agree with the numerical results of [1], where δ was up to 0.5. Their relevance to the experimental parameter range requires further numerical study. The paper is organized as follows. In Section we study energy minimizing breathers and show existence, and symmetry and monotonicity properties. In Section 3 we show the existence of shelf-like solutions for the local and nonlocal nonlinearities, in Subsections 3.1, 3. respectively. In Section 4 we study the spectra of the linearization around different breathers. In Subsection 4.1 we study the linear stability of one-peak breather solutions in the small linear coupling regime and relate these solutions to the minimizers of Section. In Subsections 4., 4.3 we present results on the essential spectra of shelf-type solutions in the local and nonlocal cases, respectively.. Nonlocal discrete NLS and energy-minimizing breather solutions We consider the one-dimensional discrete NLS equation u n = δi(u n+1 +u n 1 u n )+γ tanh κ i( m Ze κ m n u m )u n n Z, (.1) with δ, γ, κ real constants, κ>0, see [7]. Equation (.1) can be written formally as the Hamiltonian system u n = i H u, n Z, with (.) n H = δ u n+1 u n γ tanh κ n Z u m e κ m n u n. (.3) n Zm Z The quantity P = n Z u n is formally a conserved quantity. We consider solutions of equation (.1) of the breather form u n = e iωt A n,with ω real, and A : Z C independent of t. Such A satisfies ωa n = δ(a n+1 +A n 1 A n )+γ tanh κ ( m Ze κ m n A m )A n, n Z. (.4)

4 146 BREATHERS OF NONLOCAL DISCRETE NLS Let p denote the l p -norm for complex-valued functions on Z. Let Y = l (Z,C) with the norm =. Let {ê n } n Z denote the standard basis in Y. Let λ>0 and define I λ =inf{h(v): v Y ),P(v)=λ}. (.5) A configuration A Y, P (A )=λ, satisfying H(A )=I λ is referred to as a minimizer or ground state (of H at power P = λ). The set of minimizers of H at power λ is denoted as M λ. For q Z, τ q A denotes the translation of A by q, i.e. (τ q A) n = A n q. The operation of multiplying A Y by a scalar e iφ, φ R, is referred to a global phase change. Translations and global phase changes leave P and H invariant. Theorem.1. Let δ, γ>0. Then for every λ>0, thesetm λ is nonempty, and A M λ implies that A satisfies property (.4). Proof. We will only present a sketch of the proof. The existence part uses a discrete version of Lyon s concentration-compactness principle, following [0,, 30, 3]. The functional H is easily seen to be bounded below, and has the form H = δh γv, with H, V homogeneous quadratic and quartic functionals respectively. Both H (u) andv (u) arepositiveforanyu 0 in X. The discrete version of the concentration-compactness lemma (see [], Sections 3, 4) states that there are three scenarios: splitting, vanishing, and convergence (up to translation), indicated by the number Γ in [], (3.13). The goal is to show convergence, for any λ>0. To eliminate splitting, we use the subadditivity property (Lemma 4.3, []), and a discrete analogue of Lemma 4.4 of [3]. These arguments are valid for all λ>0. To eliminate vanishing for a given λ>0 it suffices to find a test function f X, withp (f)=λ, andh(f) < 0. We use functions Then P (f)=λ, and the condition H(f) < 0 becomes f = Cg, g X, C = λ g 1. (.6) δ Dg g <λ, (.7) γv (g) where (Dg) n = g n+1 g n, n Z. We use test functions (g) n = e α n, n Z, withα>0 (as in [3]) to see that Dg g =+O(α), V(g) g 4 4 = 1 α +O(1), as α 0+. (.8) Then, given any λ>0, the left hand side of inequality (.7) can be made smaller by taking α>0 sufficiently small. It follows that a minimizing sequence will converge up to translations and global phase rotations, and that M λ is nonempty for all λ>0. We easily check that H : Y R is C 1,andthatH (A) ωp (A) = 0 implies that A is a solution of equation (.4). Theorem.. Let δ, γ>0, and consider λ>0. Then A M λ implies that there exist φ R, q Z such that A = e iφ τ q A,withAreal and positive, reflection symmetric and non-increasing, i.e. A n 0, n Z, A n = A n, n Z, anda a A b, for all positive integers a<b. We first define a symmetrization operation that transforms any u Y = l (Z,C) into a configuration w Y that is real, non-negative and symmetric with respect to the origin,

5 ROBERTO I. BEN, JUAN PABLO BORGNA, AND PANAYOTIS PANAYOTAROS 147 and that moreover has the same l -norm and a smaller value of H. The symmetrization, and the proof of Lemma.1 below were suggested to us by Diego Rial. For f Y, define f : R C by f(x)= n Zf n e inx, x R. (.9) f is clearly π periodic. By the L theory for Fourier series, f belongs to L (S 1,C), with S 1 the interval [0,π] with0,π identified. Moreover equation (.9) defines an isomorphism between Y and L (S 1,C). The inverse is given by f n = 1 π f(x)e inx dx, n Z, (.10) π 0 i.e. f is the inverse Fourier transform of f. Moreover, we have Parseval s identity f n = 1 π f(x) dx. (.11) S 1 n Z Consider u Y,withũ as in equation (.9), and define v, w as follows: v n = 1 ũ(x) e inx dx, n Z, (.1) π S 1 and w n = v n, n Z. (.13) It follows from identity (.11) thatv, w belong to Y,andthat w = v = u. Also, by definition (.1) wehavev n = v n, n Z. Then definition (.13) implies that w n = w n 0, n Z. (.14) The configuration w is therefore real, positive, and symmetric, with w = u. Lemma.1. Consider u Y,andv, w Y as in definitions (.1) and(.13). Then, H(w) H(v) H(u). Proof. First we examine the quartic part V of H in Fourier space. Letting g n = e κ n, n Z, we compute g(x)= n Ze κ n e inx = 1 e κ 1 e κ. (.15) cosx+e κ We observe that g(x) = g(x), x R. Also, V (u)= u n g n m u m (.16) n Zm Z = 1 8π 3 π π π g(x)ũ(y)ũ (y +x)ũ(z)ũ (z x) dx dy dz (.17)

6 148 BREATHERS OF NONLOCAL DISCRETE NLS 1 π π π 8π 3 g(x) ũ(y) ũ (y +x) ũ(z) ũ (z x) dx dy dz. (.18) The definition of v in equation (.1) and the isomorphism between Y and L (S 1,C) imply ũ(x) =ṽ(x), a.e. x S 1, so that the R.H.S. of inequality (.18)isV(v). Therefore V (u) V (v). (.19) On the other hand, by the definition of w in equation (.13) wealsohavev (v)=v (w), hence For the quadratic part H we calculate H (u)= 1 π π since ũ(x) =ṽ(x), a.e. x S 1.Then 0 V (u) V (w). (.0) sin x ũ(x) dx = H (v), (.1) H (v)= v n+1 v n n+1 v n n R n R v = H (w), (.) using the definition of w in equation (.13). Combining inequalities (.0) and(.) we conclude that H(u) H(w), as stated. We also observe that by equation (.1) v 0 = 1 π π Then equation (.13) implies 0 ũ(x) dx 0; v n v 0, n Z\{0}. (.3) w n w 0, n Z\{0}, (.4) i.e. the maximum of w is attained at the origin. The second operation that decreases H while keeping the l norm fixed is that of replacing u Y by its non-increasing rearrangement. To define this operation, let S be the set of all f : Z R that satisfy (i) f n = f n, n>0, (ii) f 0 > 0, (iii)f 0 f n 0, n>0, and (iv) f n 0as n. Note that S is the union of its disjoint subsets S +, S 0,andS F,wheref S + implies f n > 0, n Z, f S 0 implies that there exist n for which f n = 0 and that f n 0 for an infinite subset of n Z, andf S F implies that the set of n for which f n 0 is finite. For f S + S 0 as above, let f j the j-th largest value of f, starting from j =1, i.e. f 1 = f 0,andj Z + = {1,,...}, for all f S. For if f S F,thenf attains a finite number T (f) of values; then f j, j T(f) isthej-th largest value of f, starting from j = 1, and we set f j = 0, for all j>t (f). Given a subset J of Z, χ J will denote the characteristic function of J. Wealsouse the abbreviated notation χ f a to denote the characteristic function of the set of n Z satisfying f n a. Note that the support of χ f f j is finite, for any f S,andj 1. In what follows χ J (n) denotes the value of χ J at site n.

7 ROBERTO I. BEN, JUAN PABLO BORGNA, AND PANAYOTIS PANAYOTAROS 149 Definition.1. Let I S be the set of finite I Z satisfying 0 I, andn I implies n I. Let m be such that I =m+1. Then I denotes the set of m+1 consecutive integers from m to m. Let f S + S F,fixF j > 0andletI = {n Z : f n f j } = suppχ f f j. Then I I S and we let χ f f j = χ I. For f S we define its symmetric non-increasing rearrangement as the function f : Z R + {0} defined by f = (f j f j+1 ) χ f f j. (.5) j=1 We check that f S + S F,andthatf m f n for all m>n 0. We recall the following additional facts. Proposition.1. If f S then g n = (f j f j+1 ) χ f f j (n), n Z, (.6) j=1 coincides with f, i.e. g n = f n, n Z. Ifalsof S l 1,then n = n Zf f n. (.7) n Z Remark.1. The second conclusion of Proposition.1 also implies that h l p = h l p, for all h S l p, p>0. This follows by setting f n =(h n ) p. Proof. First, suppose that n is such that f n = f m,withf j > 0. Then χ f f j (n)=1, if j m, and vanishes otherwise, and therefore g n = (f j f j+1 ) = lim (f m f N )=f m = f n, (.8) N j=m since f j 0asj. In the case where f n = 0 for some n, theng n =0 by equation (.5), since n is not in the support of any χ f f j with f j > 0. For the second statement, we first consider the case f S +. Let N (f j ), j Z + be the set of n satisfying f n = f j.then(f j ) are finite, mutually disjoint, and their union is Z. Moreover the set of f j, j Z + is the set of values of f. Then f n = f j N (f j ), (.9) n Z moreover N (f j ) = suppχ f f j. Therefore equation (.9) implies n Z j=1 j=1 f n = f j χ f f j = f j χ f f j, (.30) since the support of χ f f j belongs to I S. We check that f j = f j,andthatχ f f j = χ f f j, j Z +. Thus equation (.30) implies j=1 f n = f j χ j f f = n Z j=1 j=1f j N (f j )= f n. (.31) n Z

8 150 BREATHERS OF NONLOCAL DISCRETE NLS The case f S 0 is similar. We note there that the set where f n = 0 is not included in the sum in equation (.9), i.e. we only sum over the positive values of f. The set of positive values of f is precisely the set of the f j, j Z +, and the remaining steps are identical since the sets N (f j ) satisfy the same properties. In the case f S F we only sum over the values f j, j T(f) in equation (.9). The remaining steps are identical as the N (f j ), 1 j T(f), satisfy the same properties. From the properties of the configuration w Y obtained from u Y via equations (.1) and(.13), as above, it is clear that w S l, and we can apply the operation of non-increasing rearrangement to obtain a new symmetric configuration w that is nonincreasing, and satisfies w = w. To show that H(w) H(w), we use Lemma. below on rearrangements of quadratic forms. The lemma is a special case of a discrete analogue of the Riesz rearrangement inequality, see e.g. [17]. Let k be a positive integer and let I, J be finite subsets of Z, andlet Q(I,k,J)= χ I (n)h k (m n)χ J (m), (.3) n Zm Z where H k (n)=1 if n k, and vanishes otherwise. Lemma.. Let I, J I S, and consider the corresponding I, J, as in Definition.1. Assume also that I J. Then for any positive integer k Q(I,k,J) Q(I,k,J). (.33) Given a pair I, J the inequality is strict for at least one positive k, unless I = I, J = J. Proof. Fix a positive integer k. Consider first the case I = J. Let1 j 1 <j,...,< j m denote the positive sites of J. Then J =m+1. A gap G of J is a symmetric set of consecutive sites j not in J that satisfy either j λ <j<j λ+1 or j λ+1 <j< j λ for a pair of 0 <j λ <j λ+1 in J. Note that the definition implies that g = j λ+1 j λ > 1, and that G consists of g>0 sites. A gap is specified by the pair of sites j λ, j λ+1 above, referred to below as its (positive) edges. If J = J, thenj has no gaps. The idea is to show that deleting gaps in I and J simultaneously increases Q. We then generalize to the case I J. Given a gap G specified by the sites j λ, j λ+1 of J as above, let JsubG I S be thesetwhosepositiveelementsareallj J that satisfy j j λ, and all j g, j J that satisfy j j λ +g +1. It is easy to show that Q(JsubG,k,JsubG) Q(J,k,J). Iterating the elimination of gaps we obtain Q(J,k,J) Q(J,k,J). In the case I J, the definition of gaps is the same. We first eliminate all gaps of J whose positive sites are to the right of the largest site of I. The resulting set is denoted as J 1, and we easily argue that Q(I,k,J 1 ) Q(I,k,J). Also I J 1, moreover the remaining gaps of J 1 are also gaps of I, otherwise there would be sites of I not in J 1. Eliminating all the gaps of J 1 from both I and J 1 we obtain new sets I,andJ respectively. We have J = J, and we easily show that Q(I,k,J) Q(I,k,J 1 ). Also I J. If I J, theni still has gaps, and we eliminate them to obtain I. We again can show that Q(I,k,J) Q(I,k,J). The above hold for any positive integer k. It is easy to show in each of the above steps, that cancellation of any gap would give strict inequality for all values of k 1. For instance, the relation I J at all stages implies that the elimination of the common points in the gap will increase for each i in the smaller set the number of j in the larger set satisfying i j k. The increase will be strict for the points at the edges of the gap, for all k 1.

9 ROBERTO I. BEN, JUAN PABLO BORGNA, AND PANAYOTIS PANAYOTAROS 151 Lemma.3. Consider the configuration w S Y, and its non-increasing symmetrization w Y, as above. Then H(w) H(w). The inequality is strict, unless w = w. Proof. We first consider the quartic part V, using the functions f n = wn, f n = w n, n Z, i.e. f, f S l 1.Alsoletg(n)=e κ n, n Z. WecanuseProposition.1 and the definition H k after equation (.3) towriteg = j=1 (gj g j+1 )H j. Then, V (u)= f n g(n m)f m n Zm Z (.34) = [ a j1 χ f f j 1 (n)][ b j H j (m n)][ a j χ f f j (m)], n Zm Z j 1=1 j=1 j =1 (.35) with a j = f j f j+1, b j = g j g j+1. The order of summations can be changed using the l summability of w, andthe fact that convolution with g defines a bounded operator in Y (we omit therefore the steps of defining all quantities for finite sums and taking limits). We then have ] V (u)= = j 1=1 j=1 [ j1 b j a j χ f f j (n)h j (m n)χ 1 f f j (m) j =1a n Zm Z (.36) a j1 b j a j Q(suppχ f f j 1,j,suppχ f f j ). (.37) j 1=1 j=1 j =1 Since suppχ f f j suppχ f f j if f j f j, we can apply Lemma. for the supports of the χ f f j to obtain from equation (.37) V (f) j 1=1 j=1 j =1 The inequality is strict unless a j1 b j a j Q(suppχ f f j 1,j,suppχ f f j ) (.38) = V (f). (.39) Q(suppχ f f j 1,j,suppχ f f j )=Q(suppχ f f j 1,j,suppχ f f j ), (.40) for all j 1, j. In that case we see that w = w. The quadratic part H can be written as H (w)= n Z (w n+1 +w n w n w n+1 )= w n Zw n w n+1, (.41) and we define H + by H + (w)= n Z (w n+1 +w n +w n w n+1 )= w + n Zw n w n+1. (.4) To show that H (w) H (w) it then suffices to show that H + (w) H+ (w). We write H + (w)= w n M(n m)w m, (.43) n Zm Z

10 15 BREATHERS OF NONLOCAL DISCRETE NLS with M(0) =, M(±1) = 1, and M(n)=0 for n Z\{ 1,0,1}. M is symmetric and decreasing in Z +, and we can then follow the argument used for V above to see that H + (w) H+ (w), as required. It follows immediately by H(w)=H (w) V (w) H (w) V (w)=h(w), with strict inequality if w w, as stated. Remark.. In the case δ, γ<0 we have a similar statement for the maximizer of H, i.e. we apply the theorem to the minimizer of H. Remark.3. For δ>0, γ<0 and any λ>0wehavei λ = 0. We can see that by considering the functions g N, N Z +,wheregn N = λ/ (N +1) if N n N, and vanishes elsewhere. It follows that the infimum is not attained and that we can not use minimization to find critical points. In the case of δ sufficiently small critical points were obtained by different arguments, see [1]. Remark.4. The symmetrization and rearrangement results above also apply to energy minimizing breathers of the cubic DNLS in Z, where we minimize H q = δh γh 4, H 4 (u)= u 4 l, over configurations u Y with P (u)=λ>0. Theexistenceofa 4 minimizer for arbitrary λ>0isshownin[3]. Lemma.1 also holds for H 4,usingthe k limit g(x)=1, x, in equation (.15). Lemma.3 clearly applies to H q as well. 3. Infinite shelf-type breathers In this section we show the existence of infinite shelf-type breather solutions of equation (.4)with δ 0 and small. We consider two cases depending on the interaction term in equation (.1): local case and nonlocal case, corresponding to κ =, and κ>0 respectively. In Theorem 3.1 we show the continuation of shelf-type breathers of equation (.4) withδ = 0 in the local case. Theorem 3. concerns the continuation of shelf-type solutions for arbitrary κ> Local case. Let ω R. For A l (Z,R), δ R and n Z define F n : l (Z,R) R R by F n (A,δ)=δ(ΔA) n +γa 3 n +ωa n, (3.1) where (ΔA) n = A n 1 A n +A n+1. Let α>0 and define S 0 = {n 0 +1,n 0 +,n 0 +3,...} Z, S A = Z\S 0. We consider a nontrivial solution A = Ã of F n(a,0) = 0, n Z, obtained by requiring A n 0 for n S A,andA n =0 for n S 0.Asδ = 0, the equation F n = 0 becomes γ A n = ω, n S A. (3.) Then γ and ω must have opposite signs and we consider the solution with α = ω γ. Ã n = { 0, if n S0, α, if n S A, (3.3) Remark 3.1. We can obtain more nontrivial solutions by taking A n < 0 for all n S A. Moreover, we can choose A n 0forn S A := S + S, with A n > 0forn S + and A n < 0 for n S provided S + is finite or S is finite. The arguments below apply to these solutions as well. Let X = l (Z,R) with the norm = X as in Section. Let Z,Y denote the operator norm of a linear operator from Z to Y, both Banach spaces.

11 ROBERTO I. BEN, JUAN PABLO BORGNA, AND PANAYOTIS PANAYOTAROS 153 We define G : X R X as G(B,δ)={G n (B,δ)} n Z = F n (B +Ã,δ), (3.4) with F n given by equation (3.1) It can be checked that G is well-defined, i.e. if B X, δ R, theng(b,δ) X. Theorem 3.1. Let G be as in definition (3.4) and consider the solution (x 0,y 0 )=(0,0) of G(B,δ)=0. Then there exists ɛ 0 > 0 and a unique continuous function B :( ɛ 0,ɛ 0 ) X satisfying B(0) = 0 and G(B(δ),δ)=0 for all δ ( ɛ 0,ɛ 0 ). The proof uses the implicit function theorem and is similar to that of Theorem 3. below (with simpler computations) and is omitted. 3.. Nonlocal case. We now show existence results for arbitrary κ>0and δ small enough. Let ω R and κ>0. For A l (Z,R) andδ R define F n,n Z, by ( F n (A,δ)=δ(ΔA n )+γtanh κ e κ m n A m )A n +ωa n. (3.5) We consider, as before, a nontrivial solution A = Ã of F n(a,0) = 0, n Z, determined by α>0, and sets S 0 = {n 0 +1,n 0 +,n 0 +3,...} Z, S A = Z\S 0, for which A n > 0if n S A,andA n =0 if n S 0. Evaluating δ = 0 in equation (3.5), the equation F n =0 becomes γ tanh κ e κ m n A m = ω, n S A. (3.6) m S A We also write equation (3.6) as ( MJ = γ tanh κ ) 1 ωε, (3.7) where J m = A m, ε =[...,1,1,1] T R SA and M is defined implicitly by the previous equation. Letting ρ = e κ we have M and its inverse M 1 explicitly as ρ ρ ρ 3 M = ρ 1 ρ ρ ρ ρ 1 ρ, 1+ρ ρ 0 0 M 1 = 1 1 ρ ρ 1+ρ ρ 0 0 ρ 1+ρ ρ. (3.8)... ρ 3 ρ ρ ρ 1 The calculation of M 1 used the fact that M is Toeplitz. It is clear from equation (3.8) that M 1 is also bounded in X. Hence the vector J is computed explicitly as ( J = γ tanh κ ) 1 ωm 1 ε, where M 1 ε = 1 ρ [ ] T 1...,1,1,1,. 1+ρ 1 ρ m Z Letting α = ( γ tanh κ ) 1 ω 1 ρ 1+ρ,thesolutionÃdetermined by α>0is α, if n S A,n<n 0 Ã n = α 1 ρ, if n S A,n= n 0, 0, if n S 0. (3.9)

12 154 BREATHERS OF NONLOCAL DISCRETE NLS In a similar way we can find nontrivial solutions such that A n < 0 for all n S A. Moreover, as in Remark 3.1, we can find nontrivial solutions such that A n 0 for n S A := S + S,withA n > 0forn S + and A n < 0forn S provided S + is finite or S is finite. The arguments below are valid for these solutions as well. Then we define G : X R X as G(B,δ)={G n (B,δ)} n Z = {F n (B +Ã,δ)} n Z. (3.10) The fact that G is well-defined follows from the arguments in the proof of Lemma 3.1 below. Theorem 3.. Let G be as in equation (3.10) and fix ω, κ>0. Letn 0 be a positive integer, and consider a nontrivial solution (Ã,0) of G(B,δ)=0 with S A = {...,n 0,n 0 1,n 0 }. Then there exists ɛ 0 > 0 and a unique continuous function B :( ɛ 0,ɛ 0 ) X satisfying B(0) = 0 and G(B(δ),δ)=0 for all δ ( ɛ 0,ɛ 0 ). Proof. As in the previous theorem, we apply the implicit function theorem around asolution(ã,0) of G(B,δ) = 0. We present here the (Fréchet) derivative D 1G and we show that it is a linear isomorphism of X. The continuity of G and D 1 G at (0,0) is proved in Lemmas 3.1 and 3. below. We have ( ) M1 0 D 1 G(0,0) =, (3.11) 0 M with M 1 =cα ρ ρ ρ3 1 ρ ρ 1 ρ ρ ρ 1 ρ 3 ρ 1 ρ 1 ρ ρ 1 ρ ρ 1 ρ ρ 1 ρ 1 1 ρ, M = ω 1 ρ ρ ρ , and ρ = e κ, c =γtanh κ. As M is an infinite diagonal matrix with 1 ρ M (n,n) < 1, n Z, toprovethat D 1 G is a linear isomorphism in X it is enough to show that M 1 is invertible. We see that M 1 1 = with M 1 1 clearly bounded in X. 1 (cα )(1 ρ ) ρ ρ 0 0 ρ 1+ρ ρ 0 0 ρ 1+ρ ρ 1 ρ 1 ρ ρ 1 ρ Remark 3.. Observe that the nonlocal case approaches the local case as κ, and ρ = e κ 0. Remark 3.3. We note that [9] propose a discrete nonlinear diffusion equation whose static front solutions satisfy equation (3.6) with the term (1 ω)a n +A n added to the right hand side (one may assume A n > 0 for all n S A ). The numerical results of [9] suggest that this variant of equation (3.6) also has a solution with a maximum at the interface.,

13 ROBERTO I. BEN, JUAN PABLO BORGNA, AND PANAYOTIS PANAYOTAROS 155 To prove the continuity of G and D 1 G at (0,0) we will need to use the following discrete version of Young s inequality: Theorem 3.3 (Young s inequality, [17], ch. 4). Let p, q, r 1, 1 p + 1 q > 1 and 1 p + 1 q = 1+ 1 r. Then u lp,andv l q imply u v l r and u v r u p v q. (3.1) The inequality follows from the statement for functions on the line, see e.g. [17], ch. 4. Lemma 3.1. Let G be as in equation (3.10) and fix ω, andκ>0. ThenG is continuous at (0,0) X R. Proof. Let (B,δ) X R, then G(B,δ) G(0,0) X = F n (B +Ã,δ) Fn(Ã,0) n Z = δδ(b n +Ãn)+c γ,κ (B n +Ãn) e κ m n B m +Ãm n Z m Z ( ) +ω(b n +Ãn) c γ,κ m Ze κ m n à m à n ωãn, (3.13) with c γ,κ =γtanh κ,and C n = m Ze ( ) κ m n B m +Ãm (B n +Ãn) à mãn, n Z. (3.14) Then G(B,δ) G(0,0) X 6 ( δ Δ X,X B X + ω B X + γ tanhκ C X ). (3.15) The first and second term in the last inequality vanish as δ 0andB 0 X. It is then enough to show that C X vanishes as δ 0andB 0 X. Wehave with C X = ( ) ρ m n B m +Ãm (B n +Ãn) à mãn (3.16) n Z m Z = n Z m Zρ m n B m +Ãm B n + ( ρ m n B m +Ãm à m )Ãn (3.17) m Z n Z m Zρ m n B m +Ãm B n + ρ m n B m (B m +Ãm)Ãn (3.18) m Z Γ( B, A ) n Z m Zρ m n B n + ρ m n B m, (3.19) m Z Γ( B, A )=4( B + A ) + A ( B +4 A ). (3.0)

14 156 BREATHERS OF NONLOCAL DISCRETE NLS Therefore C X Γ( B, A ) B n ρ m n + ρ m n B m (3.1) n Z m Z n S A m Z ( ) Γ( B, A ) B X + ρ B (3.) ( X Γ( B, A ) B X + ρ 1 B X ). (3.3) The last inequality follows from Young s inequality (3.1) withu = ρ, v = B, r =, p = 1 and q =. By equation (3.0) and inequality (3.3), C X tends to zero if B X tends to zero. Lemma 3.. Fix ω and κ>0. and consider the function G : X R X defined in (3.10). Then D 1 G is continuous at (Ã,0). Proof. It is enough to show that there exists β such that [D 1 G(B,δ) D 1 G(0,0)]v β v, v X, (3.4) and β 0as(B,δ) 0 X R. WehavealreadydeterminedD 1 G(0,0) in the proof of 3.1. We now calculate D 1 G(B,δ) withb 0, and δ 0, which (we check) is given by its partial derivatives. For n Z we have ( G n (B,δ)= δ +γtanh κ ρ m n B m +Ãm +(B n +Ãn) )+ω, (3.5) B n m Z G n (B,δ)=δ +4γtanh κ ( ) ρ m n B m +Ãm (B n +Ãn), m {(n 1),(n+1)}, B m (3.6) G n (B,δ)=4γtanh κ ( ) ρ m n B m +Ãm (B n +Ãn), m Z\{(n 1),n,(n+1)}. B m (3.7) Let v X and define M = D 1 G(B,δ) D 1 G(0,0), w = Mv. Then for every n Z we have w n = M n,m v m = ( Gn (B,δ) G ) n (0,0) v m (3.8) B m B m m Z m Z = δ(δv) n +γtanh κ ( ρ m n B m +Ãm Ãm ) v n (3.9) m Z +4γtanh κ ( ) ρ m n B m +Ãm (B n +Ãn) Ãm Ãn v m (3.30) m Z = δ(δv) n + γ tanh κ (K n +J n ), (3.31) where K n = m Zρ ( m n B m +Ãm Ãm )v n, n Z, (3.3)

15 ROBERTO I. BEN, JUAN PABLO BORGNA, AND PANAYOTIS PANAYOTAROS 157 J n = m Zρ ( ) m n B m +Ãm (B n +Ãn) Ãm Ãn v m, n Z. (3.33) Hence w X δ Δ X X v X + γ tanh κ K X +4 γ tanh κ J X. (3.34) Observe that by equation (3.3) K n m Zρ ( ) m n Bm + B m Ãm v n, so that K X ( ) ρ m n Bm + B m Ãm v n (3.35) n Z m Z n Z B ( B + Ã ) ρ m n v n (3.36) m Z B ( B + Ã ) ρ v X, (3.37) 1 using Young s inequality (3.1) withu = ρ, r =, p = 1 and q =. On the other hand, by equation (3.33) observe that J n m Zρ ( ) m n B m B n + B m Ãn + B n Ãm v m, (3.38) so that J X ( ) ρ m n B m B n + B m Ãn + B n Ãm v m (3.39) n Z m Z n Z B ( B + Ã ) ρ m n v m (3.40) m Z B ( B + Ã ) ρ v X, (3.41) using again Young s inequality. Using inequalities (3.35) and(3.40) in inequality (3.34) wehavethat w ( δ Δ X X +6 γ tanh κ ) B ( B + Ã ) ρ 1 v. (3.4) By equation (3.9) and inequality (3.4) wethushaveinequality(3.4) withβ that vanishes as (B,δ) 0 X R, as required. 4. Spectral analysis of 1-peak and shelf-type breathers In this section we present results on the spectra of the linearization around some of the breather solutions considered in the previous sections and in [1]. We will consider the one-peak breather solutions, related to the minimizers of Section, and shelf-type breather of the local and nonlocal equation, studied in Section 3. 1

16 158 BREATHERS OF NONLOCAL DISCRETE NLS The linear stability analysis uses the fact that breather solutions are relative equilibria of equation (.1) with respect to the action of global phase change. Using the variable v defined by u = e iωt v, Hamilton s equation (.) becomes v n = i H ω vn, n Z, with H ω = H ωp. (4.1) A breather solution u = e iωt A of equation (.1) isafixedpointv = A of equation (4.1). Let z =[q,p] T X X, withz n =[q n,p n ] T, q n =Rev n, p n =Imv n, n Z. Thenequation (4.1) is also written as ż = J h ω, with h ω = 1 H ω, (4.) and (Jz) n = [p n,q n ] T, i.e. J is the standard symplectic operator in X X. The linearization at a fixed point A of equation (4.1) is ż = JHz, with H = h ω (A), (4.3) i.e. H is the Hessian of h ω at A. To simplify the notation we set γ = 1 in this section. This does not affect the generality of the results. We calculate JH explicitly as [ ] 0 I J =, I 0 [ ] L+ 0 H =, 0 L (4.4) where L = ωi δδ+a, L + = ωi δδ+a+4m, (4.5) and A, M are linear operators in X defined by A(n,k)=tanh κ ( m Ze κ m n A m)δ n,k, n,κ Z, (4.6) M(n,k)=tanh κ e κ n k A k A n, n,k Z, (4.7) with δ n,k the Kronecker delta. The linearization around breathers of the local cubic DNLS is obtained by taking the limit κ in equations (4.6) and(4.7). The above calculations are formal. In the case A l (Z,R), the operators A, M, L, L + are bounded operators in X, while J, H, andjh are bounded operators in X X. We also see that L, L + and H are symmetric. Similar statements apply to the local case. Let X = l (Z,R) with the inner product (u,v)= n Z u nv n. The norm is denoted by.letx c be the complexification of X, with the inner product (u,v) c = n Z u nv n. Let Y = l (Z,C) viewed as a real space with the (real) inner product u,v = Z [(Reu n)(rev n )+(Imu n )(Imv n )]. Let Y c = l (Z,C ) with the inner product u,v c = Z [(Reu n)(rev n ) +(Imu n )(Imv n ) ]. Y c is the complexification of Y. Note that Y = X X, andy c = X c X c. The notation for X, Y is that of the previous sections, where we only used the Banach space structure.

17 ROBERTO I. BEN, JUAN PABLO BORGNA, AND PANAYOTIS PANAYOTAROS 159 Let M be a bounded operator in Y c,orx c. We denote the spectrum of M by σ(m). The essential spectrum σ e (M) ofm is defined as in [14], p.9, namely λ σ(m) belongs to σ e (M) ifm λi either fails to be Fredholm, or is Fredholm with nonzero index. The definition of [15], p.43, requires a weaker condition, namely M λi not semi-fredholm. Let X be a real or complex Hilbert space, with norm X,andlet{ε j } j=1 be an orthonormal basis of X. A bounded operator T : X X is Hilbert Schmidt in X if j=1 Tε j X < (this is independent of the choice of the basis). Recall that a Hilbert Schmidt operator is compact Stability of one-peak breathers for small linear coupling. We will consider the one-peak breather solutions, defined below. These are solutions of equation (.4) witha X, and start by showing some facts on arbitrary such solutions. Lemma 4.1. Let A X. Then the operators A, M defined in equations (4.6), (4.7) respectively are Hilbert Schmidt in X, X c. Proof. Weusethebasis{ê n } n Z of X and X c. Both A, M map X X c to X, and it is enough to show the statement for X. First, by definition (4.6) n Z Aê n = tanh κ e κ m n A m n Z m Z = tanh κ g A l, (4.8) where g n = e κ n,and(a ) n = A n, n Z. By the discrete Young s inequality (3.1) we have g A l g l 1 A l. (4.9) Since A l A, both quantities are finite, and equation (4.8) and inequality (4.9) imply that A is Hilbert Schmidt in X. Similarly, by definition (4.7) wehave λ Z Mê λ = tanh κ therefore M is also Hilbert Schmidt inx. A λ λ Z n Z e κ λ n A n A 4 l, (4.10) Lemma 4.1 is applied to fixed points A X of equation (4.1). Define H 0 as [ ] ωi δδ 0 H 0 =. (4.11) 0 ωi δδ Clearly H 0 is a bounded operator in Y c, moreover JH is a compact perturbation of JH 0 : Proposition 4.1. Let A X, andletjh : Y Y be as in equations (4.4)-(4.7). Then (i) σ e (JH)=σ e (JH 0 ), and (ii) σ e (JH 0 ) consists of z C with Rez =0,Imz [ ω, ω + 4δ] [ω 4δ,ω], forδ>0, andrez =0,Imz [ ω +4δ, ω] [ω,ω 4δ], ifδ<0. Proof. (Proof of Part (i).) To show that σ e (JH)=σ e (JH 0 ) it is enough to show that JH JH 0 is Hilbert Schmidt in Y.Wehave [ ] 0 A JH JH 0 =. (4.1) A+4M 0

18 160 BREATHERS OF NONLOCAL DISCRETE NLS The Hilbert Schmidt property in Y = X X follows easily from Lemma 4.1. We can use the basis vectors ê j n, n Z, j =1,, defined as ê 1 n(k)=δ n,k [1,0] T,ê n(k)=δ n,k [0,1] T, for all n, k Z. Part (ii) is shown in the next subsection. We now consider the stability of the 1 peak solution studied numerically in [1]. Consider a solution A X of equation (.4) withδ = 0 that satisfies A n = 0, for all n Z\{n 0 },anda n0 > 0. The parameter κ>0 is fixed and arbitrary. Such a solution is unique, and by Theorem.4 of [1] it can be continued uniquely to a C 0 family of solutions A(δ), δ ( δ 0,δ 0 ), of equation (.4). We refer to these solutions as 1-peak breathers. Remark 4.1. Note that the continuation result in [1] is for solutions with A(δ) X fixed. Then ω is also a function of δ, and is not known a priori. The continuation result for fixed ω has a very similar proof, this is seen in the previous section. For the δ = 0 breather A(0), JH is block diagonal with blocks, each block corresponding to an integer k Z. The block corresponding to k = n 0 has a double eigenvalue λ n0 = 0. The other blocks have eigenvalues ±λ k = ±iω(1 e κ k n0 ), k Z\{n 0 }, (4.13) with corresponding eigenfunctions w ± k Y c,givenby w ± k (n)=δ k,n[1, i] T, n Z. (4.14) Thus σ(jh) consists of a double zero eigenvalue, an infinite number of pairs of imaginary eigenvalues ±λ k, κ>n 0, of multiplicity, and their accumulation points ±iω (belonging to σ e (JH)). Considering now the linearized operator JH corresponding to a 1-peak breather A(δ), we have by Proposition 4.1 that σ e (JH)=σ(JH 0 ). The persistence of the double zero eigenvalue is a well-known fact, it follows from the fact that L A(δ) = 0, the continuity of finite rank spectral projections, and the symmetry of spectra of symplectic operators. We now consider the persistence of the internal mode eigenvalues λ k of equation (4.13). The linearized operator (4.4)-(4.7) corresponding to the 1-peak breather A(δ), δ ( δ 0,δ 0 ), will be denoted by JH(δ). Proposition 4.. Let k>n 0, and consider the pair of imaginary eigenvalues ±λ = ±λ k of the JH(0), asinequation(4.13). Then there exists a δ 0,k > 0 for which the operators JH(δ):Y Y, δ ( δ 0,k,δ 0,k ), have two eigenvalues ±λ j (δ) ir, j =1,. The families ±λ j (δ), j =1, are continuous in δ and satisfy ±λ j (0) = ±λ. Proof. Fix k>n 0. By the continuity of finite dimensional spectral projections and the corresponding eigenvalues under perturbations by bounded operators, see [15], the families ±λ j (δ), j =1, are continuous in δ, with±λ(0) = ±λ = ±λ k, as stated, assuming that δ belongstosomeinterval( δ 0,k,δ 0,k ). Consider the eigenvalue λ(0) of JH(0). By the symmetry of spectra of symplectic operators the two continued eigenvalues of JH(δ) are either on the imaginary axis, or are off the imaginary axis and satisfy λ (δ)= λ 1(δ). To eliminate the second scenario we use the Krein dichotomy argument, see [14], p. 181, (7.14). In particular, if the λ j (δ) have nonzero real parts then the corresponding eigenvectors v j (δ) mustsatisfy H(δ)v j (δ),v j (δ) c =0, δ ( δ 0,k,δ 0,k ), j=1,. (4.15)

19 ROBERTO I. BEN, JUAN PABLO BORGNA, AND PANAYOTIS PANAYOTAROS 161 On the other hand, using explicit expressions for the eigenvectors v j (0), j =1,, i.e. v 1 (0) = w + k, v (0) = w + m,withm<n 0 satisfying n 0 m = k n 0,andw + k, w+ m as in equation (4.14), we compute that H(0)v j (0),v j (0) c = ω(1 e κ k n0 ) 0, j=1,, (4.16) contradicting equation (4.15) atδ =0. We expect that for δ, γ>0, δ sufficiently small, the 1 peak breathers obtained by continuation should coincide with the global minimizers of Section. Nevertheless at the moment we can prove the following weaker statement. Proposition 4.3. Let λ>0, and fix n 0 Z. Consider the continuous one-parameter curve of A(δ) X, δ [0,δ 0 ], of solutions of equation (.4) witha(0) the1-peakbreather solution supported at n = n 0,and A(δ) = λ, δ [0,δ 0 ]. Then there is a sequence of {δ m } m Z + [0,δ 0 ], that converges to δ =0, and such that A(δ m ) minimizes H of (.3) over all configurations u X with P (u)=λ. Proof. Fix λ>0. Let H δ = H = δh H 4,whereH, V are the quadratic, and quartic parts of H respectively, see equation (.3). For each δ [0,δ 0 ], let u δ X be a minimizer of H δ at power P = λ. Forδ, δ [0,δ 0 ], we have that δ<δ implies H δ (u δ ) H δ (u δ ) H δ (u δ ). Also, we easily check that δ δ implies H δ (u δ ) H δ (u δ ). It follows that any sequence of points u δ, δ [0,δ 0 ], that converges to u 0 is a minimizing sequence for H 0 = V, and therefore has a convergent subsequence, up to translations, by the arguments of Theorem.1. On the other hand, the critical points of V are solutions of equation (3.6). The support of minimizers with finite support consists of sets of consecutive sites by Theorem.. Such solutions were described in [1], where we saw that all solutions with support consisting of more that three sites must have their maxima at interfaces. Then the only solution having the properties of Theorem. is the one supported on one site. We also observe that by Young s inequality, the right hand side of equation (3.6) maps X to X. Thus there are no solutions with infinite support. The points of the minimizing subsequence are solutions of equation (.4), and must therefore come arbitrarily close to the breather A(0), after suitable translations. By the uniqueness of the continuation, Theorem.4, [1], these points must eventually belong to the branch A(δ) of solutions obtained for some interval [0,δ 0 ]. 4.. Essential spectrum of linearization around shelf-type breather: local case. We now consider the essential spectrum of the linearization around the shelf solutions of Section 3. We first consider the shelf solutions of the local (power) nonlinearity system. The main result is Theorem 4.1 below. It follows from an analysis of the piecewise linear discrete dynamical system defined by the linear operator. We give a fairly complete presentation of the theory. In the nonlocal case we have a weaker result, Theorem 4. below, using similar computations. The theory however is not complete. In the case of the power nonlinearity system, we consider solutions obtained by continuation of the δ = 0 breather { α, if n 0, Ã n = (4.17) 0, if n 1, with α = ω. This is a special case of the configuration of solution (3.3). The spectral results below are the same for the general case. For ω fixed, let B n, n Z, be a solution of Theorem 3.1 and let A n = Ãn +B n, n Z. (4.18)

20 16 BREATHERS OF NONLOCAL DISCRETE NLS Let L,δ = JH, withjh as in equation (4.4), L, L + as in equation (4.5), A, M as in equations (4.6) and(4.7) witha n as in equation (4.18), à n as in equation (4.17), n Z. OperatorL,δ is therefore the formal linearization around the shelf solution of Theorem 3.1. Also let L = JH, withjh as in equation (4.4), with L, L + as in equation (4.5), and A(n,k)=A 0 (n,k)=ã nδ n,k, M(n,k)=M 0 (n,k)=ã nδ n,k, (4.19) where Ãn is as in equation (4.17). Proposition 4.4. The operator L,δ is a compact perturbation of L, and therefore σ e (L,δ )=σ e (L) in Y. The prove the statement it is enough to show that A A 0, M M 0 are Hilbert Schmidt. The proof of this fact is a simplified version of the proof of Lemma 4.5 for the nonlocal problem below, and is omitted. To calculate the essential spectrum of L we study the equation (L λ)z =0, z = [q,p] T,withz n =[q n,p n ] T as a discrete dynamical system. The (spectral) parameter λ varies over C. By the definition of L, witha, M of equation (4.19), and à of equation (4.17) wehavethatlz = λz is equivalent to δ(p n+1 +p n 1 p n )=λq n, ωq n +δ(q n+1 +q n 1 q n )=λp n, n 0; (4.0) ωp n δ(p n+1 +p n 1 p n )=λq n, ωq n +δ(q n+1 +q n 1 q n )=λp n, n 1. (4.1) Finite difference equation (4.0), (4.1) is furthermore equivalent to a first order system y n+1 = F(y n ) for a sequence of vectors y n =[y n (1),...,y n (4)] T C 4, n Z. This formulation uses the identification [y n (1),y n (),y n (3),y n (4)] T =[q n 1,q n,p n 1,p n ] T. (4.) By equation (4.), n 0, equation (4.0) is then equivalent to y n+1 = M (λ)y n, n 0, with M (λ)= 1 ω δ + 0 λ δ (4.3) 0 λ δ 1 Similarly by equation (4.), n 1, equation (4.1) is equivalent to y n+1 = M + (λ)y n, n 1, with M + (λ)= 1 ω δ + 0 λ δ (4.4) 0 λ δ 1 ω δ + Equations (4.3)and(4.4) define a piecewise linear discrete-time dynamical system that may be written as y n+1 = M(λ)y n, where M(λ)=M (λ) ifn 0, and M(λ)= M + (λ) ifn 1. Proposition 4.5. Given any m Z, a C 4, there exists a unique y n, n Z, satisfying equations (4.3) and(4.4) andy m = a.

21 ROBERTO I. BEN, JUAN PABLO BORGNA, AND PANAYOTIS PANAYOTAROS 163 Proof. We check first that the M ± (λ) are invertible, λ C. By equations (4.3) and (4.4) P (r;λ)=det(m (λ) ri)=[(r 1) r ω δ ](r 1) +r λ δ, (4.5) P + (r;λ)=det(m + (λ) ri)=[ r( ω δ r)+1] +r λ δ. (4.6) We see that P ± (r,λ)=0, r = 0 leads to a contradiction, λ C. Thus the M ± (λ) are invertible, λ C. This fact allows us to iterate conditions (4.3) and(4.4) backwards as well, writing [M (λ)] 1 y n+1 = y n, n 0, (4.7) [M + (λ)] 1 y n+1 = y n, n 1. (4.8) We can thus iterate both forwards and backwards in each segment. We check that equations (4.3), (4.4), (4.7), and (4.8) allow us to cross the interface separating the M ± (λ) segments in a unique way, in either direction. We examine the dynamics of each of the systems (4.3) and(4.4), particularly its dependence on the parameter λ. In both cases we can start with some y 1 and consider the iteration backwards, and forwards respectively. Since each system is linear this amounts to a spectral analysis of the matrices M ± (λ). The results of this analysis are then interpreted in operator-analytic terms for the resolvent equation (L λ)z = f, f =[g,h] T X X Y c. Consider a linear system x n+1 = Tx n in C N,whereT an invertible N N matrix, and n belongs to a set of indices J that is either Z, or the set of all integers above or below some m Z. The system is hyperbolic if there exist subspaces E s, E u of C N, dim(e s )+dim(e u )=N that are invariant under T, and for which there exist C>0, ϱ>1 such that T n P s v C N Cϱ n P s v C N, T n P u v C N Cϱ n P u v C N, n J, (4.9) where P s, P u are respective projections to E s, E u (and commute with T ). The subspaces E s, E u are referred to as stable and unstable subspaces respectively. Hyperbolicity can be often verified by calculating the eigenvalues of T. The dimensions of E s, E u are determined be the multiplicities of eigenvalues inside and outside the unit circle respectively. In the present problem hyperbolicity will be examined for each of the systems (4.3) and (4.4). The corresponding stable and unstable subspaces will be denoted by E±, s E±. u Note that the E, s E u for system (4.3) refer to the forward iteration of M (λ). Lemmas 4., 4.3 connect the analysis of the eigenvalues of the M ± (λ) totheof spectrum of L. The proofs are at the end of the subsection. Lemma 4.. Let λ C be such that the systems (4.3) and(4.4) defined by M (λ), M + (λ) respectively are both hyperbolic, with dim(e )+dim(e u +)=N, s E u E+ s = {0}. Then λ belongs to the residual set of L in Y. Lemma 4.3. Let λ C, and suppose that one of M (λ), M + (λ) has a semisimple eigenvalue on the unit circle. Then λ belongs to the essential spectrum of L in Y.