Bifurcations and stability of gap solitons in periodic potentials

Transcription

1 PHYSICAL REVIEW E 70, (2004) Bifurcations an stability of gap solitons in perioic potentials Dmitry E. Pelinovsky, 1 Anrey A. Sukhorukov, 2 an Yuri S. Kivshar 2 1 Department of Mathematics, McMaster University, Hamilton, Ontario, Canaa L8S 4K1 2 Nonlinear Physics Group an Centre for Ultra-high Banwith Devices for Optical Systems (CUDOS), Research School of Physical Sciences an Engineering, Australian National University, Canberra, ACT 0200, Australia (Receive 10 May 2004; publishe 30 September 2004) We analyze the existence, stability, an internal moes of gap solitons in nonlinear perioic systems escribe by the nonlinear Schröinger equation with a sinusoial potential, such as photonic crystals, waveguie arrays, optically-inuce photonic lattices, an Bose-Einstein conensates loae onto an optical lattice. We stuy bifurcations of gap solitons from the ban eges of the Floquet-Bloch spectrum, an show that gap solitons can appear near all lower or upper ban eges of the spectrum, for focusing or efocusing nonlinearity, respectively. We show that, in general, two types of gap solitons can bifurcate from each ban ege, an one of those two is always unstable. A gap soliton corresponing to a given ban ege is shown to possess a number of internal moes that bifurcate from all ban eges of the same polarity. We emonstrate that stability of gap solitons is etermine by location of the internal moes with respect to the spectral bans of the inverte spectrum an, when they overlap, complex eigenvalues give rise to oscillatory instabilities of gap solitons. DOI: /PhysRevE PACS number(s): Tg, Jx, Qs, Lm I. INTRODUCTION Perioic structures are very common in physical problems, with the crystalline lattice being the most familiar classical example. One of the important features of such systems is the existence of multiple frequency gaps in the wave transmission spectra. Such spectral gaps are responsible for a strong moification of the wave ispersion an iffraction that occurs when waves experience resonant Bragg scattering from a perioic structure [1]. When nonlinear self-action becomes important, the systems with perioically moulate parameters emonstrate a number of new effects; in particular they can support a novel type of solitons, the so-calle gap solitons, which exist in the gaps of the linear wave spectrum ue to a strong Bragg scattering an coupling between the forwar an backwar propagating moes [2]. During the last years, it was shown that gap solitons may exist in ifferent types of nonlinear perioic structures incluing low-imensional photonic crystals an photonic layere structures [3,4], waveguie arrays [5], optically inuce photonic lattices [6,7], an Bose- Einstein conensates loae onto an optical lattice [8,9]. There are known two simplifie approaches to stuy nonlinear localize moes an gap solitons in perioic structures [10]. The first approach is base on the erivation of an effective iscrete nonlinear Schröinger equation an the analysis of its stationary localize solutions in the form of iscrete localize moes or iscrete solitons [11]. In the soli-state physics, the similar approach is known as the tight-bining approximation that, in application to perioic photonic structures, correspons to the case of weakly couple efect moes excite in each iniviual waveguie of the structure. The analogous concepts appear in the stuy of other systems such as the Bose-Einstein conensates in optical lattices [12]. *URL: On the other han, weak nonlinear effects in optical fibers with a perioic moulation of the refractive inex are well stuie in the framework of the other familiar an wellaccepte approach, the couple-moe theory [2]. The couple-moe theory for nonlinear perioic structures is base on a ecomposition of the wave fiel into the forwar an backwar propagating moes, uner the conition of the Bragg resonance, an the erivation of a system of linearly couple nonlinear equations for those two moes. Such an approach is usually applie to analyze nonlinear localize waves in the systems with a weakly moulate optical refractive inex known as gap (or Bragg) solitons. A number of recent experiments in the nonlinear guie wave optics [5 7] an Bose-Einstein conensates [9] were conucte in the perioic structures uner the conitions when those approximations may be invali. Inee, one of the main features of the wave propagation in perioic structures is the existence of a set of multiple forbien gaps in the transmission spectrum. As a result, the nonlinearly inuce localization of waves can become possible in each of these gaps. However, the effective iscrete equation erive in the tight-bining approximation escribes only one transmission ban surroune by two semi-infinite ban gaps an, therefore, a fine structure of the ban-gap spectrum associate with the wave transmission in perioic meia is lost in this approach. On the other han, the couple-moe theory of gap solitons escribes only an isolate narrow gap in between two semi-infinite transmission bans, an it oes not allow one to consier simultaneously the localize moes ue to the total internal reflection as well as to stuy the ban coupling an interban resonances. Recently, it was realize that the stuy of the simultaneous existence of localize moes of ifferent types is a very important issue in the analysis of stability of nonlinear localize moes an gap solitons [13]. The main purpose of this paper is to analyze the existence, bifurcations, an stability of spatially localize nonlinear moes (i.e., lattice an gap solitons) in the framework of /2004/70(3)/036618(17)/$ The American Physical Society

2 PELINOVSKY, SUKHORUKOV, AND KIVSHAR PHYSICAL REVIEW E 70, (2004) an effective continuous moel escribe by the nonlinear Schröinger equation with a perioic external potential. The use of this well-accepte nonlinear moel for our analysis allows us to remove all restrictions of both approaches mentione above, an to stuy consistently the effects of the ban-gap spectrum on the properties an linear stability of gap solitons. First, by applying the multiscale asymptotic analytical methos, we show that such gap solitons may appear in all ban gaps of the perioic potential for any sign of nonlinearity, but they bifurcate from ifferent ban eges for ifferent signs of nonlinearity. Secon, we emonstrate that, in general, only two branches of gap solitons bifurcate from each ban ege, an one of those two is always linearly unstable. Thir, we stuy stability of gap solitons in a selecte ban gap an fin the soliton internal moes bifurcating from all other ban eges of the same polarity [14,15]. However, only one internal moe can bifurcate from the ban ege where the gap soliton originates itself. At last, we analyze the conitions when the bifurcation of the internal moes can give rise to complex eigenvalues, which are shown to be responsible for oscillatory instabilities of gap solitons [16]. The paper is organize as follows. Section II presents our physical moel which is escribe by an effective nonlinear Schröinger equation with an external perioic potential of the simplest sinusoial form. Section III summarizes the stuies of the spectral properties of the linear eigenvalue problem with a perioic potential. In Sec. IV we stuy bifurcations of gap solitons by means of the weakly nonlinear approximation. Section V presents the analysis of the exponentially small corrections beyon the weakly nonlinear approximation. Section VI iscusses the stability problem of gap solitons. Symmetry-breaking instabilities are stuie in Sec. VII. Internal an oscillatory instability moes of gap solitons associate with nonzero eigenvalues are stuie in Sec. VIII. Finally, Sec. IX summarizes our results an iscusses further perspectives. Appenix A gives etails of the numerical metho for calculations of eigenvalues. Appenixes B an C present etails of erivations, which are use in Secs. VII an VIII, respectively. II. MODEL We consier the cubic nonlinear Schröinger (NLS) equation with an external perioic potential in the form, i t = xx Vx 2, where Vx=Vx, is the funamental perio, an = ±1 efines the type of the wave self-action effect, namely self-focusing =1 or self-efocusing =1. The analytical results presente below are rather general, an they are vali for ifferent types of smooth arbitrary-shape perioic potentials. However, in the numerical examples iscusse below we consier the square sine potential, 1 x Vx = V 0 sin2. 2 The harmonic potential (2) escribes, in the mean-fiel approximation, the ynamics of the Bose-Einstein conensate in an optical lattice, when the parabolic trap is neglecte [8,9]. The square sine potential Vx has two extremum points on the perio of x, such that x=0 is the minimum an x=/2 is the maximum of Vx. Stationary localize solutions of the cubic NLS equation (1) for gap solitons are foun in the form x,t=xexp it, where is referre to as the soliton parameter. The soliton profile x is foun as a spatially localize solution of the nonlinear problem: Vx 2 =, 3 where the prime stans for a erivative in x. Existence an multiplicity of multihumpe localize states x in the spectral gaps of the perioic potential Vx were consiere by means of the variational methos by Alama an Li [17,18]. Bifurcations of boun states were analyze by Kupper an Stuart [19,20] an Heinz an Stuart [21] who prove that, epening on the sign of the nonlinear term, lower or upper enpoints of the continuous spectrum are bifurcation points. Extension to the multiimensional case was evelope with Bloch waves of the linear Schröinger operator [22]. The number of branches of boun states was classifie in terms of the eigenvalues of the linear Schröinger operator with perioic an ecaying potentials [23]. Eigenvalues of the latter (linear) problem were previously consiere by Gesztesy et al. [24] an Alama et al. [25]. In application to the problem of the Bose-Einstein conensates in optical lattices, the stationary moel (3) has been consiere recently by Louis et al. [8] who foun numerically ifferent types of spatially localize solutions in ifferent ban gaps. All previous results were restricte to the stuy of the existence of spatially localize solutions. Here, we use more general methos (but, in some sense, less rigorous from the mathematical point of view) an stuy bifurcations, stability, an internal moes of gap solitons. To achieve these objectives, we apply the multiscale perturbation series expansion methos, evelope earlier by Iizuka [26] an Iizuka an Waati [27]. With the perturbation series methos, we classify systematically the branches of gap solitons bifurcating from the ban eges to the ban gaps, as well as their stability. III. SPECTRAL BANDS AND GAPS Perioic potential Vx inuces a ban-gap structure in the linear Schröinger spectral problem: Vx =. 4 The spectral bans are locate for ban, where we enumerate the ban eges in the following orer: ban = 0, 1 3, 2 4, 5 7, 6. 5 The spectral bans are compute for the square sine potential (2) an the results are shown in Fig. 1. A review of

3 BIFURCATIONS AND STABILITY OF GAP SOLITONS PHYSICAL REVIEW E 70, (2004) FIG. 1. The structure of spectral bans of the linear perioic problem (4): (a) Trace of funamental matrix vs ; (b) Floquet exponent k vs ; an (c) Soli: Bloch waves at the ban eges, as inicate by arrows; Dashe: potential Vx. Parameters are V 0 =1 an =10. spectral theory for perioic potentials can be foun in the book by Eastham [28]. Here we recover some etails which are important for our analysis. When the spectral parameter is taken insie the spectral bans, i.e., ban, the problem (4) has two linearly inepenent solutions in the form of Bloch waves, 1 = 1 xe ikx, 2 = 2 xe ikx, 6 where 1,2 x are perioic functions an k is the Floquet exponent, which can be chosen insie the first Brillouin zone such that 0k/. The graph of k is shown in Fig. 1(b) for the first three spectral bans. The spectral bans of the perioic potential Vx are escribe by the function, which is the trace of the funamental matrix of solutions [28] = 2 cos k. The spectral bans are efine for 22, which correspons to propagating waves with real k. On the other han, the waves become exponentially localize insie the gaps, where 2 an Imk0. A characteristic epenence is isplaye in Fig. 1(a). There are infinitely many spectral bans for a oneimensional perioic potential Vx, where 2 [28]. If n 0 at the ban ege = n, two ajacent spectral bans o not overlap, such that the corresponing ban gap has a nonzero with. We consier the nonegenerate spectral ban, such that n 0 at the en point = n. The even-numbere ban eges = 2m, m0 correspon to perioic Bloch functions 2m x= 2m x, while the o-numbere ban eges = 2m1, m1 correspon to antiperioic Bloch functions 2m1 x= 2m1 x. The Bloch functions n x for the first five ban eges = 0, 1, 3, 2, 4 are shown in Fig. 1(c). We now emonstrate that the bifurcations of boun states an stationary gap solitons may occur when the two funamental solutions 1,2 x in Eq. (6) become linearly epenent. Since 1 xe 2ikx solves the same equation as 2 x but it is not a perioic function of x, unless k 7 =0 mo2/ or k=/mo2/, the two solutions 1,2 x are always linearly inepenent in the interior of the spectral bans ban. On the other han, the two solutions 1,2 x become linearly epenent at the ban eges = n, since 2m x = 1 x = 2 x 8 an k 2m =0 mo2/ at the even-numbere ban eges, an 2m1 x = 1 xe ix/ = 2 xe ix/ 9 an k 2m1 =/mo2/ at the o-numbere ban eges. The ban ege = n has geometric multiplicity one with the only linearly inepenent Bloch function n x. The secon, linearly inepenent solution of Eq. (4) at = n grows linearly in x. The ban ege = n has, however, algebraic multiplicity two, since there exists a generalize Bloch function 1 n x that solves the erivative problem: 1 n Vx 1 n n 1 n =2 n x. 10 It follows from Eq. (10) that the generalize Bloch functions 1 1 2m x an 2m1 x are perioic an antiperioic in x, respectively. We conclue that the ban eges = n are the only bifurcation points of the linear spectrum ban, associate with the perioic potential Vx. The ban curvature near the ban ege = n follows from the expansion of efine in Eq. (7): n n n O n 2 = 1 n 2 2 k k n 2 Ok k n 4, 11 where k 2m =0 an k 2m1 =. As a result, n =21 n an such that = n n 2 k k n 2 Ok k n 4, n 2 = 2 1 n n

4 PELINOVSKY, SUKHORUKOV, AND KIVSHAR PHYSICAL REVIEW E 70, (2004) 2 The ban curvatures n can be expresse in terms of Bloch functions n x an 1 n x at = n. We use perturbation series expansions for funamental solutions 1,2 x near the ban eges = n : 1,2 xe ±ik n x = n x ± ik k n 1 n x k k n 2 2 n x Ok k n The eigenvalue is expane in the perturbation series (12). The secon-orer correction 2 n x satisfies the nonhomogeneous linear equation: 2 n Vx 2 n n 2 n =2 1 n 1 2 n n. 14 If 1,2 x are perioic functions of x, the secon-orer correction 2 n x in the perturbation series (13) is perioic for n=2m an antiperioic for n=2m1. By Freholm Alternative, this conition is satisfie if the right-han-sie of Eq. (14) satisfies the constraint: 1 2 n n 0 2 x n n 20 1 x =0. 15 Therefore the ban curvature 2 n is expresse in terms of integrals of n x an 1 n x. The perturbation series expansions (12) an (13) can be continue algorithmically to the higher orers in powers of kk n. IV. BIFURCATIONS OF GAP SOLITONS Nonlinear boun states (gap solitons) of the NLS equation (1) are stationary solutions in the form: x,t = s xe i s t, 16 where the real function s x ecays to zero as x an satisfies the nonlinear problem, s Vx s s 3 = s s. 17 When s x is small, the nonlinear potential s 2 x acts as a perturbation term to the perioic potential Vx. The perturbation term leas to bifurcation of gap solitons s x from the ban eges = n of the linear ban-gap spectrum. We stuy bifurcations of gap solitons with the multiscale perturbation series expansions: an where s = n 2 n O 4 18 s x = x;x, X = x x 0, 1, 19 x;x = A n X n x A n X 1 n x 2 2 s x;x O Here A n X is a space-ecaying boun state an n x is the perioic or antiperioic Bloch function. Parameter x 0 etermines a location of A n X with respect to n x. The Bloch functions n x an n 1 x are efine from the linear problems (4) an (10). The secon-orer correction term s 2 x;x satisfies the linear non-homogeneous equation: s 2 Vx s 2 n s 2 = A n n 2 n 1 A n 3 n 3 n A n n. 21 The secular growth of s 2 x;x in x is remove if the righthan-sie of (21) satisfies the Freholm conition, which reuces to the nonlinear equation for A n =A n X: where n 2 A n n 2 A n 3 n A n =0, n 2 = 0 0 n 4 x n 2 x Using the constraint (22), we represent the secon-orer correction term s 2 x;x in the form: s 2 x;x = A n X n 2 x A n 3 X n nl2 x, 24 where n 2 x solves the nonhomogeneous problem (14), while n nl2 x solves the problem, nl2 n Vx nl2 n n nl2 n = 2 n n 3 n. 25 The nonlinear equation (22) is just the stationary NLS equation, which has sech-solitons if sgn 2 n =sgn 2 n =sgn n. For the focusing nonlinearity, =1, the sechsolitons bifurcate from all ban eges, where 2 n 0, such that n 0. It follows from Eq. (12) that branches of gap solitons etach from all lower ban eges ownwars from the corresponing ban gaps. For the efocusing nonlinearity, = 1, the sech-solitons bifurcate from all ban eges, where 2 n 0, such that n 0. Therefore branches of gap solitons etach from all upper ban eges upwars from the corresponing ban gaps. Branches of gap solitons are shown in Fig. 2 for =1 near the ban eges = 0, = 3, an = 4 an in Fig. 3 for = 1 near the ban eges = 1 an = 2. The families of gap solitons have been foun by solving the nonlinear eigenvalue problem (17) with a stanar relaxation technique [29]. The sech-solitons of the nonlinear equation (22) are written explicitly in the form: A n X = a n sech n X, where a n an n are foun from equations, 26 n n 2 n 2 =0, n 2 a n 2 2 n 2 n 2 =0, 27 provie that sgn n 2 =sgn n 2 =sgn n. The sech-type soliton envelopes (26) always have a single-humpe profile. Since A n X is the envelope of n x, the resulting nonlinear boun state s x has the oscillatory structure near the ban ege s = n

5 BIFURCATIONS AND STABILITY OF GAP SOLITONS PHYSICAL REVIEW E 70, (2004) FIG. 2. Bifurcations for the on-site an off-site gap solitons in a self-focusing meium (=1). Top: the soliton power P = 2 s x;x vs. Soli: solitons centere at x 0 =0, ashe: centere at x 0 =/2. (a f): spatial profiles of gap solitons corresponing to marke points in the upper plot; shaing marks the minima of the potential Vx. V. BRANCHES OF GAP SOLITONS DUE TO SYMMETRY BREAKING The absence of translational invariance along the x irection, associate with the presence of the perioic potential, has an important effect on the soliton properties. For example, it was foun that iscrete solitons, bifurcating from the first ban, can be centere at (on-site) or in-between (offsite) potential wells. In this section, we emonstrate that two branches of gap solitons, bifurcating from all the bans, are centere at ifferent positions in the perioic potential. The gap soliton s x near the ban ege s = n is represente by the perturbation series expansions (18) (20), provie that the formal series converges. Parameter x 0 in the slow coorinate X=xx 0 etermines the location of the boun state A n X with respect to the Bloch function n x. We will show that only two values of x 0 on the perio of x secure convergence of the formal series, in the general case. Our analysis is equivalent to the construction of the Melnikov function, which gives the istance between separatrices in the nonlinear oscillator with a small, rapily varying force [30,31]. Zeros of the Melnikov function inicate values of x 0, where the separatrices intersect, so that a homoclinic orbit for the gap soliton exists in the nonlinear problem (17) with the perioic potential Vx. We will erive the Melnikov function [30,31] with a simple but equivalent metho. Derivative of the nonlinear FIG. 3. Bifurcations for the on-site an off-site gap solitons in a self-efocusing meium =1. Notations are the same as in Fig

6 PELINOVSKY, SUKHORUKOV, AND KIVSHAR PHYSICAL REVIEW E 70, (2004) equation (17) in x results in the following thir-orer orinary ifferential equation: Fˆ k = FXe ikx X. 36 s Vx s s s 3 s 2 s Vx s =0. 28 If the gap soliton s x exists, then it satisfies zero bounary conitions as x. Multiplication of Eq. (28) by s x an integration over x result in the following constraint: M s x 0 Vx s = 2 xx =0. 29 The function M s x 0 is the Melnikov function for the existence of homoclinic orbits [30,31]. The constraint (29) is always satisfie if the gap soliton s x an the potential Vx are symmetric with respect to the location of the central peak at x=x 0, such that 2 s xx 0 = 2 s x 0 x an Vxx 0 =Vx 0 x. More precise information on the constraint (29) can be obtaine near the ban ege s = n, where the perturbation series expansions (18) (20) are vali. The function x;x has the power series expansion in, each term of which satisfies the square-perioic bounary conitions in x, 2 x ;X = 2 x;x, an the ecaying bounary conitions in X, 30 lim x;x =0. 31 X We shall prove that M s x 0 is exponentially small in terms of. To o so, we rewrite Eq. (28) for x;x, multiply it by x;x, an integrate the resulting equation over x0,. Using the perioic bounary conition (30), we erive the relation, 0 Vx 2 x;xx =2 X0,x 2 x 2 2,x,X x. 32 X0 Using the ecaying bounary conition (31), we prove that Vx X0 2 x;xx =0. 33 As a result, the function Vx 2 x;x is expane in Fourier series in x as Vx 2 x;x = m= such that W n,m X;=W n,m X; an W n,m X;e 2imx/, W n,0 X;X = at any orer of. The Fourier transform of FX is efine by the stanar integral: The Melnikov function M s x 0 is then expane with the use of the Fourier series (34) an the Fourier transform (36) in the form: M s x 0 = m= Ŵ n,m 2m ;e2imx 0 /. 37 At the leaing orer, we have W n,m X;0=A 2 n Xw 0 n,m, where 0 are coefficients in the Fourier series, w n,m Vx n 2 x = m= w 0 n,m e 2imx/. 38 The zero-orer term (m=0) in the series (37) is zero at any orer of, since the constraint (35) results in the conition: Ŵ n,0 0;=0. The higher-orer terms with larger values of m are exponentially smaller compare to the terms with smaller values of m in the limit 0, since  n 2 k is exponentially ecaying in k. Therefore, using exponential asymptotics, we truncate the series (37) by the first-orer terms (m=±1) in the limit 0: where an M s x 0 = 1 cos 2x 0 1 =2w 0 n,1  n 2 2 E 1 =O 2  n 2 2 argw 0 n,1 E 1, O n Assuming that 1 0, we conclue from Eq. (39) that there are precisely two families of gap solitons bifurcating from two roots of the function cos2x 0 / on the perio of x 0. We now prove that, for the square sine potential (2), the values of argw 0 n,1 are the same for all ban eges as argw 0 n,1 =/2. It is clear from Eq. (2) that Vx=Vx an Vx0 for 0x /2, while all Bloch wave square amplitues are symmetric, such that 2 n x= 2 n x. As a result, it follows from Eq. (38) that argw 0 n,1 =/2 an the two roots of x 0 occur at extremal points of Vx: x 0 =0 an x 0 =/2. The former (minimum) point correspons to the on-site gap soliton, while the latter (maximum) point correspons to the off-site gap soliton, in accorance with Figs. 2 an 3. When 1 =0 an Ŵ n,1 2/;0, higher powers of are generally nonzero in the first-orer terms m=±1, such that only two branches of gap solitons s x bifurcate in a general case. If the potential Vx is special such that Ŵ n,1 2/;=0 at any orer of but Ŵ n,2 4/;0, the leaing-orer terms in the series (37) become seconorer (m=±2), such that four branches of gap solitons s x

7 BIFURCATIONS AND STABILITY OF GAP SOLITONS PHYSICAL REVIEW E 70, (2004) may bifurcate from four roots of the function cos4x 0 / on the perio of x 0. We o not know whether any special potentials Vx may exist to hol the constraint Ŵ n,1 2/; =0 at any orer of. VI. LINEAR STABILITY OF GAP SOLITONS Stability of solitons with respect to perturbations is an important problem for applications. Stable states act as attractors, an their excitation is weakly sensitive to noise or perturbations. On the other han, unstable states ten to unergo ynamical transformations ue to a rapi growth of initial perturbations, an this behavior may be useful, for example, for switching applications [32]. We stuy the stability of gap solitons s x by consiering the evolution of perturbe solution in the following form: x,t = e i s t s x ux iwxe t ux iwxe t. 40 We substitute Eq. (40) into the NLS equation (1) an perform its linearization with respect to the functions u,w escribing small-amplitue perturbations. Then, we obtain couple linear eigenmoe equations where u,w is an eigenvector an is an eigenvalue, L 1 u =w, L 0 w = u. 41 Here L 0 an L 1 are Schröinger operators with perioic an ecaying potentials, L 0 = 2 x 2 Vx s s 2 x, 42 L 1 = 2 x 2 Vx s 3 2 s x. 43 We are intereste in eigenvalues, which correspon to the spatially localize eigenvectors u,w in L 2 R,C 2. If there exists an eigenvalue with Re0, the gap soliton s x is spectrally unstable. On the contrary, if all eigenvalues have Re=0, the gap soliton is neutrally stable. Neutral stability can result in spectral instability ue to resonances, embee eigenvalues, an bifurcations of isolate eigenvalues with Re=0. We have use an approach base on the Evans function for numerical calculation of the eigenvalues, the etails are presente in Appenix A. The stability problem (41) is written in terms of two Schröinger operators L 0 an L 1 with perioic Vx an ecaying 2 s x potentials. At the ban ege = n, where s x0, the two Schröinger operators coincie with the operator L s : L s = 2 x 2 Vx s. 44 For w=iu an =i, the spectral bans of the stability problem (41) occur at s ban, i.e., at 0 s, 1 s 3 s, 2 s... For the gap soliton bifurcating from the upper ban ege s = n, the parameter s satisfies the inequality: n s n2, while for the gap soliton bifurcating from the lower ban ege s = n, the parameter s satisfies the inequality: n2 s n. We emonstrate below that an important value which efines many stability properties is the energy of the spectral ban, which is efine by h = u,l 1 u w,l 0 w, 45 where the inner prouct, is efine for perioic Bloch functions on the perio x0,: f,g =0 fxgxx. 46 It is clear that h m =2 m n m, m at =0, where h m refers to the mth ban ege in the spectrum of L s for s = n. All spectral bans of L s, which are lower with respect to s = n, become bans of negative energy for the gap soliton s x, while all spectral bans of L s, which are upper with respect to s = n, become bans of positive energy for the gap soliton s x. The spectrum of the stability problem (41) is ouble because of the inversion symmetry: w=iu an =i.asa result, the bans of positive an negative energies of the operators L s an L s may overlap in the couple spectrum (41) for the same values of. The spectrum of the problem (41) transforms when 0. A simple an stable transformation is a shift of spectral bans of L s an L s along the imaginary axis of to the istance s n. As a result, the origin =0 becomes isolate from the spectral bans of L s an L s for any 0. Other transformations of the spectrum are possible an may result in instabilities of gap solitons. These transformations are consiere in Secs. VII an VIII. VII. SYMMETRY-BREAKING INSTABILITY OF GAP SOLITONS In Sec. V, we have ientifie two families of on-site an off-site gap solitons, which have ifferent positions with respect to the unerlying potential. In this section, we emonstrate that one of these soliton families is unstable with respect to symmetry breaking. They ten to move across the potential an eventually transform into their stable counterparts which have a ifferent position. These results generalize the previously foun instability of off-site iscrete solitons associate with the first ban [32]. More specifically, we show that the symmetry-breaking instability of gap solitons is efine by the sign of M S x 0.If M s x 0 0, then a pair of purely imaginary eigenvalues in the stability problem (41) bifurcates from =0, an these internal moes escribe oscillations of the perturbe soliton aroun the stable position x=x 0. On the other han, if M s x 0 0, then a pair of real eigenvalues bifurcates in the problem (41) an these exponentially growing instability moes characterize soliton motion away from the unstable location x=x 0. We note that these results are vali in the vicinity of gap eges, where the eigenvalues are exponen

8 PELINOVSKY, SUKHORUKOV, AND KIVSHAR PHYSICAL REVIEW E 70, (2004) tially small in terms of the perturbation parameter. Due to the symmetry of the NLS equation (1), we have a nonempty kernel of the operator L 0 for all along the family of the gap soliton s x: L 0 x;x =0. 47 On the other han, the gap soliton s x in the asymptotic representation (19) is parameterize by x 0 in the formal power series (20) in. As a result, the kernel of the operator L 1 is nonempty at all power orers of n : L 1 U =0 n, U = x;x. 48 X The zero eigenvalue of L 1 is estroye beyon the powers of, since the gap soliton s x is not parameterize by x 0, values of which are fixe by roots of the Melnikov function (29). We show in Appenix B that the zero eigenvalue of L 1, associate with the eigenfunction U x, shifts accoring to the quaratic form: U,L 1 U = 1 2 4M S x 0, 49 where the quaratic form is efine for ecaying functions on the whole line of x: f,g fxgxx. 50 = Accoring to the stanar perturbation theory [33], the quaratic form in Eq. (49) etermines the shift of the zero eigenvalue of L 1, associate with the eigenfunction U x. When M S x 0 0, the zero eigenvalue of L 1 becomes positive, while when M S x 0 0, the zero eigenvalue of L 1 becomes negative. We show that a small negative eigenvalue of L 1 results in a small real positive eigenvalue of the stability problem (41), while a small positive eigenvalue L 1 results in a pair of small imaginary eigenvalues. A small eigenvalue =, corresponing to the eigenfunction u x, can be foun from the problem: L 1 u = 2 L 0 1 u, 51 or equivalently, from the Rayleigh quotient: 2 = u,l 1 u u,l 1 0 u. 52 The quaratic form u,l 1 0 u exists if,u =0, as follows from Eq. (47). Since,U =O n an L 1 U =O n at all power orers of n, we conclue that u x = U x E, 53 where E is exponentially small in terms of. We shall prove that u,l 1 0 u = 1 4 2, O 1 54, such that u,l 1 0 u 0 at the leaing orer. It follows from the nonlinear problem (17) that L 0 X x;x =2 x;x x As a result, we have 2 2 x;x. 55 X X,X = X,L 0 1 X 1 X,L 0 1 x. The solution of the inhomogeneous problem, 56 L 0 V = x, 57 exists at all power orers of n, since the right-han sie of Eq. (57) is orthogonal to at all power orers of n. Therefore the quaratic form U,V has a regular power series in, starting with the zero-orer term. Since X,X = 1 4 2, X X 2 x, 58 an the secon term is exponentially small in, Eq. (56) reuces to Eq. (54) at the leaing orer, such that the Rayleigh quotient (52) is given in the leaing orer by 2 2M S x 0 2,. 59 It follows from Eq. (59) that a negative eigenvalue of L 1 for M S x 0 0 results in a small positive eigenvalue in the stability problem (41). The exponentially small correction of the function M s x 0 is given by the expansion (39), where argw 0 n,1 =/2 for the square-sine potential (2). Therefore, M S x 0 0 for x 0 =0 an M S x 0 0 for x 0 = 2. In the former case, the gap soliton s x is locate at the minimum point of Vx an it has a pair of small imaginary eigenvalues. In the latter case, the gap soliton s x is locate at the maximum point of Vx an it is unstable with a small real positive eigenvalue. Figure 4 shows unstable eigenvalues, splitting from zero eigenvalues, for the branches of gap solitons with x 0 =/2. We note that stability of on-site an off-site solitons can be interchange in more complex potentials, such as binary superlattices [34]. Aitionally, stability can change eep insie the gap, where the asymptotic analysis is not applicable [35]. Asymptotic results for NLS solitons in the lowest semiinfinite ban gap in the focusing case =1 were obtaine recently by Kapitula [36] in the limit Vx 0. Branches of NLS solitons s x= NLS x= 2s sech 2s xx 0 in the small perioic potential function Vx are efine by zeros of the function M s x 0, given by Eq. (29) with s = NLS x. Stability of branches of NLS solitons is efine by the erivative M S x 0, such that the NLS solitons bifurcating from the minimum points of Vx are stable, while the NLS solitons bifurcating from the maximum points of Vx are unstable. We note that the opposite conclusion is rawn in Ref. [36], ue to an elementary sign error

9 BIFURCATIONS AND STABILITY OF GAP SOLITONS PHYSICAL REVIEW E 70, (2004) FIG. 4. Eigenvalues corresponing to symmetry-breaking instabilities of gap solitons centere at x 0 =/2 for =1 (shown with ashe lines in Fig. 2). VIII. INTERNAL MODES AND OSCILLATORY INSTABILITIES OF GAP SOLITONS Apart from symmetry-breaking instabilities analyze in the previous section, we emonstrate that gap solitons can exhibit a ifferent type, so-calle oscillatory instabilities. Such instabilities can occur ue to a resonance between the internal moes corresponing to the eges of the gap in which soliton is localize, as was emonstrate within the couple-moe equations [16]. However, the couple-moe theory escribe only an isolate ban gap, whereas it was foun that oscillatory instability can occur ue to resonance between ifferent gaps [13,34]. Such resonances can result in a resonant energy reistribution between the gaps an a formation of breathing structures, as was recently emonstrate experimentally [6]. In this section, we present a systematic analysis of such instabilities an show that they appear when a sieban associate with the intergap resonances falls outsie a ban gap. Oscillatory moes an instabilities are characterize by eigenvalues with a nonzero imaginary part of the stability problem (41) for 0. First, we show that new imaginary eigenvalues with ecaying eigenvectors u,w bifurcate from the ban eges =i m n of the same polarity as the ban ege s = n. Bifurcations of internal moes occur generally at the orer of O 2,if m n. These eigenvalues are referre to as the internal moes of gap solitons [14,15], an in our case such moes appear ue to a resonance between the gap eges m an n. Such resonances are possible because a soliton inuces an effective waveguie, which can support localize moes in other gaps [37]. In Fig. 5, we show three moes of operator L 0 supporte in the semiinfinite gap near the ege = 0 by a gap soliton existing in the gap near the ege = 2 in the case of a self-focusing nonlinearity =1. Secon, we show that resonance between internal moes of the operator L s an the bans of the inverte spectrum of L s occurs if the bifurcating internal moe of L s becomes embee into the spectral ban of L s. When it happens, embee internal moes bifurcate generally to complex eigenvalues, leaing to oscillatory instabilities of the gap soliton s x. Resonant bifurcations of complex eigenvalues occur generally at orer of O 4. Thir, we show that the internal moe of L s may occur near the ban ege of the inverte spectrum of L s. In this case, bifurcations of isolate, embee, an complex eigenvalues are all possible at the orer of O 4, epening on the configuration of the spectral bans of L s an L s. Finally, we show that at most one internal moe can bifurcate from the ban ege, which is closest to the zero eigenvalue. This bifurcation occurs generally at the orer of O 4. FIG. 5. Linear guie moes of operator L 0 in the semi-infinite ban gap for a gap soliton shown in Fig. 2(). Left: eigenvalues marke with ots (secon an thir ones are inistinguishable within the picture scale). Right: corresponing moe profiles

10 PELINOVSKY, SUKHORUKOV, AND KIVSHAR PHYSICAL REVIEW E 70, (2004) We emphasize that bifurcations of other eigenvalues may not generally occur in higher orers of, since the ban eges of the spectrum of L 1 an L 0 with s x0 o not support resonances in a generic case. Bifurcations of other eigenvalues may occur far from the limit =0, when the spectral bans of the linearize operator get aitional resonances at the ban eges or in the interior points. Bifurcations of the existing eigenvalues may also occur far from the limit =0 if the existing eigenvalues coalesce with each other or with the spectral bans. A. Nonresonant bifurcations of internal moes Let n be the inex of the ban ege s = n where the gap soliton s x bifurcates from. We consier a ifferent ban ege of the stability problem (41) with =i m n, such that mn. We assume that the mth ban ege of the spectrum of L s is locate in a ban gap of the inverte spectrum of L s, such that 2 n m ban. Using the same perturbation series expansions (18) (20), we expan solutions of the stability problem (41) in the perturbation series: u = B m X m x B m X 1 m x 2 u 2 m x,x O 3, 60 w = ib m X m x B m X 1 m x 2 w 2 m x,x O 3, 61 an = i m s 2 m O 4, 62 where the secon-orer correction terms u m 2,w m 2 solve the nonhomogeneous system: L s u m 2 s m w m 2 = B m m 2 m 1 m B m m 3A n 2 B m n 2 m, 63 potential (26). There is at least one isolate eigenvalue if sgn 2 m =sgn 2 nm =sgn m. In this case, the lowest eigenvalue an eigenfunction of the problem (65) can be foun explicitly as m = 2 n 2 m s 2 m, B m = sech s m n X, 67 where s m solves the quaratic equation s m s m 1 = n nm 2 2 m. 68 n Isolate eigenvalues m of the problem (65), when they exist, correspon to internal moes in the perturbation series (62), bifurcating in the ban gaps of the operators L s an L s from the ban ege =i m n. When sgn 2 m =sgn 2 nm, the linear problem (65) oes not have any isolate eigenvalues. Since sgn 2 nm =sgn 2 n =sgn, wenotice that all ban eges = n that support bifurcations of gap solitons in the nonlinear problem (17) support also bifurcations of internal moes in the spectrum of a selecte nth gap soliton. In the focusing case, =1, all lower ban eges generate internal moes ownwars from the corresponing ban gaps, i.e., 2 m 0 an m 0. In the efocusing case, = 1, all upper ban eges generate internal moes upwars from the corresponing ban eges, i.e., 2 m 0 an m 0. It is surprising that more than one internal moe coul be generate near the ban ege =i m n. In the case of no perioic potential Vx=0, perturbations of NLS solitons generate at most one internal moe [14,38]. On the other han, perturbations of gap solitons in the couple-moe equations may generate several internal moes an complex eigenvalues [16,39]. In the case of finite potential Vx, the number of internal moes epens on the epth of the square sech potential in the eigenvalue problem (65), which is etermine by parameters of the ban curvatures 2 n an 2 m an by the nonlinearity coefficients 2 n an 2 nm. L s w m 2 s m u m 2 = B m m 2 m 1 m B m m A 2 n B m 2 n m. 64 Uner the constraint that 2 s m ban, the seconorer corrections u 2 m,w 2 m are perioic or antiperioic functions of x, when a single Freholm conition is satisfie. The Freholm conition takes the form of the eigenvalue problem for m : where 2 m B m 2 2 nm A 2 n XB m m B m =0, 2 nm = 0 0 n 2 m 2 x m 2 x We note that Eq. (65) escribes the linear moes supporte by a soliton-inuce waveguie in other gaps. The linear problem (65) is a Schröinger equation with the solvable B. Resonant bifurcations of complex eigenvalues Accoring to the general expression (40), eigenvalues with a nonzero imaginary part escribe soliton oscillations, which are associate with the appearance of two sieban spatial frequencies Im an Im. Gap solitons are spectrally stable for small values of 0 with respect to a particular resonant oscillation if both of the siebans fall insie the gaps of the linear spectrum, whereas an oscillatory instability may arise when one sieban appears insie a linear transmission ban [13,34]. This general behavior is illustrate in Fig. 6, where the real part of the eigenvalue is nonzero inicating the presence of the oscillatory instability when the lower sieban is insie the transmission ban of the inverte spectrum. However, the instability is suppresse when the sieban moves insie the ban gap. The instability shown in Fig. 6 appears ue to a resonant coupling between a gap soliton marke in Fig. 2, an its own funamental guie moe in the first gap shown in Fig. 5. The characteristic profiles of instability moes are presente in Fig. 7. The

11 BIFURCATIONS AND STABILITY OF GAP SOLITONS PHYSICAL REVIEW E 70, (2004) FIG. 8. Evolution of a soliton perturbe by a linear moe corresponing to Fig. 7. FIG. 6. Eigenvalues corresponing to a resonance of a gap soliton (marke in Fig. 2) with its funamental guie moe in the semi-infinite ban gap. Small oscillations in Re are ue to numerical error. top row shows the perturbation uiw, which correspons to higher spatial frequency Im accoring to Eq. (40), an we inee see that this moe closely matches the guie moe profile (cf. Fig. 5) in agreement with the asymptotic expressions (60) an (61). On the other han, the bottom row of Fig. 7 shows the low-frequency component, which escribes the raiation waves emitte by the soliton when Im is insie the transmission ban. The long-live oscillating, or breathing, states are shown in Fig. 8. Similar effects may occur ue to a resonance with higher-orer guie moes, as shown in Figs One important ifference is that the associate breathing states can have ifferent symmetries for various excite moes, cf. Figs. 8 an 11. In mathematical terms, stability requires that all internal moes etaching from the ban eges =i m n resie insie the ban gaps of the inverte operator L s, an the zero eigenvalue of L 1 shifts to small imaginary eigenvalues. When an internal moe is embee into a spectral ban of the inverte operator L s, oscillatory instability of the gap soliton s x may arise for small values of 0. Embee imaginary eigenvalues are known to be structurally unstable with respect to small perturbations an, provie that their energy is opposite with respect to the energy ensity of the spectral ban, they bifurcate into complex eigenvalues [40]. By construction, resonance of internal moes of L s with spectral bans of L s is only possible if the internal moe, etaching from the ban ege =i m s, has the opposite energy signature with respect to the energy signature of the inverte spectral ban r =2 n m ban, such that =i m n =i n r. Therefore all embee imaginary eigenvalues in the linearize stability problem (41) are expecte to bifurcate to complex eigenvalues in a generic case. We prove in Appenix C that, provie that r =2 n m ban, we have FIG. 7. Profiles of linear moes corresponing to a resonance in Fig

12 PELINOVSKY, SUKHORUKOV, AND KIVSHAR PHYSICAL REVIEW E 70, (2004) FIG. 11. Evolution of a soliton perturbe by a linear moe corresponing to Fig. 10. ege =i m n are locate in the neighborhoo of the ban ege =i n k of the inverte spectrum. We assume here that m, k, an s satisfy the resonance conition within the mismatch of orer O 2 : FIG. 9. Eigenvalues corresponing to a resonance of a gap soliton (marke in Fig. 2) with its higher-orer guie moe in the semi-infinite ban gap. Re = 4 m O 5, m 0, 69 where is the eigenvalue of the bifurcating internal moe, given by Eq. (62). In a generic case, when m 0, the embee imaginary eigenvalue bifurcates to the unstable omain Re0 an leas to oscillatory instabilities of the gap soliton s x. C. Marginal bifurcations of internal moes an complex eigenvalues A marginal case between nonresonant an resonant bifurcations occurs when internal moes etaching from the ban m k 2 s = 2 mk. 70 In this marginal case, we expan the eigenvalue an the eigenfunction u,w of the linearize stability problem (41) in the moifie perturbation series, an where u = u 0 mk x;x u 1 mk x;x 2 u 2 mk x;x O 3, 71 w = iw 0 mk x;x w 1 mk x;x 2 w 2 mk x;x O 3, 72 = i m s 2 mk O 4, u 0 mk = B m X m x C k X k x, u 1 mk = B m X 1 m x C k X 1 k x, 73 FIG. 10. Profiles of linear moes corresponing to a resonance in Fig

13 BIFURCATIONS AND STABILITY OF GAP SOLITONS PHYSICAL REVIEW E 70, (2004) w 0 mk = B m X m x C k X k x, w 1 mk = B m X 1 m x C k X 1 k x The secon-orer correction terms u 2 mk,w 2 mk solve the system: L s u 2 mk s m w 2 mk = F 2, where L s w 2 mk s m u 2 mk = G 2, F 2 = B m m 2 m 1 C k k 2 k 1 mk B m m C k k mk C k k 3A n 2 n 2 B m m C k k, G 2 = B m m 2 m 1 C k k 2 k 1 mk B m m 74 C k k mk C k k A 2 n 2 n B m m C k k. Because of the resonance conition (70), the secon-orer corrections u 2 mk,w 2 mk are perioic or antiperioic functions of x if two Freholm conitions are satisfie. The two Freholm conitions take the form of a couple eigenvalue problem for mk : 2 m B m A 2 n X2 2 nm B m 2 nmk C k = mk B m, 2 k C k A 2 n X 2 nkm B m 2 2 nk C k = mk mk C k, 75 where 2 2 nm is efine in Eq. (66), while nmk is efine as 2 nmk = 0 n 2 m k x 0 m 2 x. 76 The couple eigenvalue problem (75) is not self-ajoint an therefore the eigenvalues mk coul be complex-value. We assume that sgn 2 m =sgn 2 nm such that the first equation (75) with C k 0 has at least one internal moe for sgn mk =sgn 2 m. For convenience, we consier here the efocusing case =1, when 2 nm 0 an 2 m 0. In this case, the internal moe of the first equation (75) with C k 0 exists for mk 0, while the spectral ban is locate for negative values of mk. There are two particular cases, epening on whether 2 k 0or 2 k 0. In the case 2 k 0, the secon equation (75) with B m 0 oes not have any internal moes, while the spectral ban is locate below the value mk mk. When mk 1, internal moes in the component B m for mk 0 are not affecte by the spectral ban in the component C k, since the following estimate hols for finite mk an large mk : C k = 2 nkm mk A n 2 XB m O 1 mk When the value of mk ecreases an becomes negative, all internal moes in the component B m become embee into the spectral ban in the component C k. The embee eigenvalues mk bifurcate as complex eigenvalues mk ue to the Fermi golen rule as in [40]. In the case of 2 k 0, the secon equation (75) with B m 0 has at least one internal moe for mk mk 0, while the spectral ban is locate above the value mk mk. When mk 1, all internal moes in the component B m for mk 0 an those in the component C k for mk mk 0 are locate in the gap between the two spectral bans. The internal moes in the component B m are not affecte by the spectral ban in the component C k, since C k is small accoring to the expansion (77). On the other han, the internal moes in the component C k are not affecte by the spectral ban in the component B m, since the following estimate hols for finite mk mk an large mk : B m = 2 nmk mk A n 2 XC k O 1 mk When the value mk increases an becomes positive, the gap between spectral bans isappear an all internal moes in the components B m an C k coalesce or become embee into overlapping spectral bans. In the first case, internal moes mk bifurcate as complex eigenvalues m ue to the Hamiltonian Hopf bifurcation. In the secon case, internal moes mk bifurcate as complex eigenvalues mk ue to the Fermi golen rule. Again, we have oscillatory instabilities of the gap soliton s x, emerging from all bifurcating internal moes of L s in resonance with the spectral bans of L s or vice verse. D. Internal moes near =0 The couple eigenvalue problem (75) is erive uner the resonance conition (70) between two ban eges of operators L s an L s. The resonance conition (70) is always satisfie for m = k = n an mk =2 n, when the ban ege =i m n =0 of the stability problem (41) coincies with the ban ege s = n of the gap soliton s x an n is use in Eq. (18). In this case, the couple eigenvalue problem (75) escribes the transformation of the spectrum of the problem (41) at 0, when a narrow spectral gap appears in the spectrum of the problem (41) near the origin =0. We showe in Sec. VII that a pair of real or purely imaginary eigenvalues bifurcate from =0 ue to the broken translational invariance. We will show here that another pair of internal moes may bifurcate insie the same gap from the ban eges. Contrary to the former bifurcation, which is exponentially small in, the latter bifurcation occurs generally in the orer of O 4. For the case m = k = n an mk =2 n, the system (75) can be simplifie ue to the obvious reuction: 2 2 m = k = 2 n an 2 nm = 2 nk = 2 nmk = 2 nkm = 2 n. Using the variables u n = B m C k, w n = ib m C k, 79 an n = i mk n, we transform the problem (75) to the form:

14 PELINOVSKY, SUKHORUKOV, AND KIVSHAR PHYSICAL REVIEW E 70, (2004) L n 1 u n = n w n, L n 0 w n = n u n, 81 n n where L 1 an L 0 are linear Schroinger operators with ecaying potentials: L 0 n = n 2 2 X 2 n n 2 A n 2 X, 82 L n 2 1 = 2 n X 2 n 3 2 n A 2 n X, 83 where sgn n =sgn 2 n =sgn 2 n. The linear eigenvalue problem (81) is the linearize NLS problem on the real line, associate to the sech-solitons (26). The problem has two branches of the continuous spectrum for n i, n i n,, the four-imensional null space n =0, an the resonance at the ban eges n =±i n. A small perturbation of the ecaying potentials in the problem (81) may result in the ege bifurcation of a single pair of internal moes n =±i n, such that n n, provie a certain integral criterion is satisfie [15,39]. It was shown [14,38] that the iscrete NLS equation with a small lattice step size supports bifurcations of a single pair of internal moes from the ban eges beyon the linearize NLS problem (81). In orer to stuy these bifurcations, we woul have to exten perturbation series expansions (71) (73) to the next orers an erive the O 2 corrections to the linearize NLS problem (81). This work goes beyon the scope of the present paper. We only note that there is at most one pair of internal moes bifurcating in the narrow gap near =0. IX. CONCLUSIONS We have presente a systematic analysis of the existence, bifurcations, linear stability, an internal moes of gap solitons in the framework of the nonlinear Schröinger equation with a perioic potential. This moel or its generalizations appear in a variety of physical applications incluing lowimensional photonic crystals, arrays of couple nonlinear optical waveguies, optically inuce photonic lattices, an Bose-Einstein conensates loae onto an optical lattice. In the framework of this moel, we have classifie branches of gap solitons bifurcating from the ban eges of the Floquet- Bloch spectrum by means of the multiscale perturbation series expansion metho. We have emonstrate that gap solitons can appear near all lower or upper ban eges of the spectrum for focusing or efocusing nonlinearity, respectively. We have stuie the gap-soliton internal moes an stability of gap solitons in the framework of the continuous moel with a perioic potential. We have emonstrate that the gap-soliton stability is etermine by the broken translational invariance, as well as the location of internal moes with respect to the spectral bans of the linearize spectrum. We have shown analytically an numerically that complex eigenvalues of the stability problem correspon to oscillatory instabilities of gap solitons. The analytical results presente above are rather general, an they are expecte to be vali for ifferent types of smooth arbitrary-shape perioic potentials. Although our numerical examples have been presente for the specific case of the sinusoial potential, we expect that our main results can be applie to other types of perioic potentials. ACKNOWLEDGMENTS The authors acknowlege support from the Australian Research Council. They thank Dr. Elena Ostrovskaya for useful collaboration an help at the initial stage of this project, as well as F. Gesztesy, B. Sanstee, an A. Scheel for useful references. D.P. thanks the Nonlinear Physics Group for hospitality uring his visit. APPENDIX A: NUMERICAL METHOD FOR CALCULATION OF EIGENVALUES Eigenvalues of the spectral problem (41) provie key information about the soliton stability. However, an accurate numerical calculation of complex eigenvalues escribing oscillatory instabilities of gap solitons is a nontrivial problem even in the case of a simpler system of couple-moe equations [41 43]. The reason for numerical ifficulties is that ifferent components of eigenvectors have very ifferent localization withs. For example, the moes shown in the bottom part of Figs. 7 an 10 are much broaer than the soliton with, while the moes shown in the top part of Figs. 7 an 10 have comparable with. Numerical approaches use in a number of earlier stuies [41 43] were base on the iscretization of Eq. (41), however, an accurate escription of weakly localize moes requires the use of impractically wie computational winows. It was suggeste that the eigenvalues can be calculate approximately, an then improve using a special iterative proceure [42,43]. Inour analysis, we avoi such problems by using a ifferent approach base on the Evans function formalism. This metho prove to be very effective for tracing soliton instabilities in perioic systems [44]. First, we reformulate the spectral problem (41) using a ifferent set of functions U=uiw an W=uiw, 2 U x 2 VxU 2 sx2u W = s iu, 2 W x 2 VxW 2 sx2w U = s iw. A1 The avantage of this formulation for the numerical analysis is that Eqs. (A1) become uncouple away from the soliton core as x. In these regions, solutions of Eqs. (A1) are foun in terms of Bloch functions, an they form a natural basis for representation of solutions along the whole line, Ux = U 1 x 1 U 2 x 2, Wx = W 1 x 1 W 2 x 2, A2 where ± 1,2 x are two linearly inepenent Bloch functions, foun as solutions of Eq. (4) with = s ±i an U 1,2 an W 1,2 are unknown parameters. By using the metho of varia