Linear instability of two-dimensional low-amplitude gap solitons near band edges in periodic media

Linear instability of two-dimensional low-amplitude gap solitons near band edges in periodic media

Zuoqiang Shi,¹Jiandong Wang,²Zhigang Chen,³and Jianke Yang^2,

*

1Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China

2Department of Mathematics and Statistics, University of Vermont, Burlington, Vermont 05401, USA

3Department of Physics and Astronomy, San Francisco State University, San Francisco, California 94132, USA 共Received 30 September 2008; published 5 December 2008兲

Previous work has shown that in a two-dimensional periodic medium under focusing or defocusing cubic nonlinearities, gap solitons in the form of low-amplitude and slowly modulated single-Bloch-wave packets can bifurcate out from the edges of Bloch bands. In this paper, linear stability properties of these gap solitons near band edges are determined both analytically and numerically. Through asymptotic analysis, it is shown that these gap solitons are linearly unstable if the slope of their power curve at the band edge has the opposite sign of nonlinearity共here focusing nonlinearity is said to have a positive sign, and defocusing nonlinearity to have a negative sign兲. An equivalent condition for linear instability is that the power of the gap solitons near the band edge is lower than the limit power value on the band edge. Through numerical computations of the power curves, it is found that this condition is always satisfied, thus two-dimensional gap solitons near band edges are linearly unstable. The analytical formula for the unstable eigenvalue of gap solitons near band edges is also asymptotically derived. It is shown that this unstable eigenvalue is proportional to the cubic power of the soliton’s amplitude, and it induces width instabilities of gap solitons. A comparison between this analytical eigenvalue formula and numerically computed eigenvalues shows excellent agreement.

DOI:10.1103/PhysRevA.78.063812 PACS number共s兲: 42.65.Tg, 05.45.Yv

I. INTRODUCTION

Nonlinear wave phenomena in periodic media are receiv- ing intensive studies in many branches of science and engi- neering these days. Two prominent examples are nonlinear optics and Bose-Einstein condensates. In nonlinear optics, a periodic medium can be created by sophisticated fabrication techniques关1,2兴, by laser writing关3,4兴, or by optical induc- tion techniques关5–7兴. In Bose-Einstein condensates, a peri- odic trapping potential for the condensates can be introduced by laser beams关8兴. The motivation for the study of nonlinear wave phenomena in these periodic media is that the periodic media exhibit very novel dispersion 共or diffraction兲 behaviors—most notably the appearance of band gaps inside the continuous spectrum 关1,9兴. These novel dispersion 共dif- fraction兲behaviors, when coupled with self-focusing or self- defocusing nonlinearity, give rise to new types of self- trapped localized states共gap solitons兲, which reside either in the semi-infinite band gap or higher band gaps 关5,10兴, and these gap solitons can be utilized for various applications 关8,12兴. So far, a wide variety of gap solitons have been re- ported either theoretically, or experimentally, or both. They include fundamental and vortex solitons in the semi-infinite gap 共under focusing nonlinearity兲 关2,5,13–16兴, fundamental and vortex solitons in the first gap共under defocusing nonlin- earity兲 关5,17,18兴, reduced-symmetry solitons and vortex- array solitons in the first gap 共under focusing nonlinearity兲 关19,20兴, gap-wave solitons in the first gap共under defocusing nonlinearity兲 关8,11兴, embedded soliton trains 共under either focusing or defocusing nonlinearity兲 关21兴, etc. Solitons in ring lattices and quasiperiodic lattices have been explored as

well 关22–24兴. Analytically one-dimensional 共1D兲 gap soli- tons bifurcating from edges of Bloch bands were investi- gated in 关25兴. It was found that the centers of such 1D gap solitons can only be at two locations. One location is at a potential minimum 共also called a lattice site兲, and such a soliton is referred to as an on-site soliton. The other location is between lattice sites, and such a soliton is referred to as an off-site soliton关26兴. In two dimensions, classifications of gap solitons bifurcating from edges of Bloch bands were per- formed in 关10兴. It was found that the centers of two- dimensional共2D兲gap solitons can only be at four locations:

one is on site, and the other three are off site. More impor- tantly, it was revealed that near band edges with two Bloch modes, the coupling between these Bloch modes gives rise to many new types of gap solitons such as reduced-symmetry solitons关19兴and vortex-array solitons关20兴兲. It has also been recognized that some gap solitons 共such as vortex solitons 关13,18兴兲 do not bifurcate from the edges of Bloch bands 关27,28兴.

Stability of gap solitons with respect to perturbations is an important issue, because only stable solitons are promising for experimental observations and physical applications. An important criterion for the linear stability of solitary waves is the Vakhitov-Kolokolov 共VK兲stability criterion, which says that under a certain spectral condition, sign-definite solitary waves are linearly unstable if and only if the slope of the power curve is negative关29–34兴 共a related criterion in terms of the Hamiltonian-power diagram for homogeneous media was given in关35兴兲. Most gap solitons, however, are not sign definite, thus the VK criterion does not apply. Because of that, other methods need to be developed. In one dimension, asymptotic analysis has been carried out for the linear stabil- ity of low-amplitude gap solitons near band edges关25兴. It has been shown that on-site gap solitons near band edges can be

*Corresponding author; [email protected]

linearly stable, while off-site solitons near band edges are always linearly unstable due to drift instabilities induced by translational-mode-related unstable eigenvalues 关25兴. Away from band edges, additional instabilities can also arise. In two dimensions, however, our knowledge on the stability of gap solitons is much more limited. It is well known that stability properties of solitary waves strongly depend on the number of spatial dimensions. Thus 1D stability results may not be carried over to the 2D case. Some limited stability results of gap solitons have been obtained in two dimensions recently. For instance, in the semi-infinite band gap under focusing cubic nonlinearity, it has been shown numerically that 2D on-site solitons are linearly stable away from the band edge where the power curve has a positive slope, but are linearly unstable near the band edge where the power curve has a negative slope 关13兴. This result can be readily explained analytically by the VK stability criterion since these gap solitons are sign definite 关29–34兴 共note that the propagation constant in 关13兴 was defined with the opposite sign of that in关29,33兴; in this paper we follow the definition of 关29,33兴兲. For certain sign-indefinite gap solitons, limited numerical stability analysis has been performed as well共for instance, on out-of-phase dipole solitons and vortex solitons兲 关13,36–38兴. It was found that such solitons may be stable in certain parameter regions. But stability properties of many other sign-indefinite gap solitons, especially in higher band gaps, are still unknown.

In this paper, we analytically determine the stability prop- erties of 2D gap solitons near edges of Bloch bands in a sinusoidal lattice potential under either the focusing or defo- cusing cubic 共Kerr兲 nonlinearity by asymptotic methods.

These gap solitons are low-amplitude slowly modulated single-Bloch-wave packets. We show that these solitons are linearly unstable if the slope of their power curve at the band edge has the opposite sign of the nonlinearity. Specifically, these gap solitons near a band edge are linearly unstable if the power curve has a negative slope under the focusing nonlinearity, or if the power curve has a positive slope under the defocusing nonlinearity. This result generalizes and modifies the VK stability criterion to sign-indefinite gap soli- tons, and it reveals the important role the sign of nonlinearity plays in the connection between linear stability and the power slope. An equivalent condition for linear instability of 2D gap solitons near band edges is that the powers of these solitons near band edges are lower than the limit power val- ues on the band edges. Our numerical computations of the power curves near many band edges indicate that this insta- bility condition is all satisfied for both on-site and off-site gap solitons, thus these 2D gap solitons near band edges are linearly unstable. This contrasts the one-dimensional case where on-site gap solitons near band edges can be linearly stable 关25兴. The asymptotic expression for the unstable ei- genvalue of 2D gap solitons near band edges is also derived.

We find that this unstable eigenvalue bifurcates from the eigenmode of the 2D envelope soliton’s zero eigenvalue in- duced by this envelope soliton’s variation with respect to its propagation constant. This eigenmode with zero eigenvalue for the 2D envelope soliton does not exist in one dimension, thus this unstable eigenvalue of 2D gap solitons has no coun- terpart in the 1D case. We also show that this unstable eigen-

value leads to width instabilities of 2D gap solitons, and its magnitude is proportional to the cubic power of the soliton’s amplitude. Lastly, the asymptotic expression for this unstable eigenvalue is compared with numerically computed eigen- values for both the focusing and defocusing nonlinearities, and excellent agreement is obtained. We point out that our instability results of 2D gap solitons near band edges do not conflict with experimental observations of such solitons in 关5,17,19兴because the observed solitons have high amplitudes and do not reside near band edges.

II. LOW-AMPLITUDE 2D GAP SOLITONS NEAR BAND EDGES

The mathematical model we consider is the 2D nonlinear Schrödinger共NLS兲equation with a periodic potential as fol- lows:

iU_t+U_xx+U_yy−V共x,y兲U+␴兩U兩²U= 0, 共2.1兲 whereU共x,y,t兲is a complex function, the potentialV共x,y兲is periodic inx andy共it is also called a lattice potential兲, and

␴=⫾1 is the sign of nonlinearity. When ␴= 1, the nonlin- earity is of self-focusing type, while when ␴= −1, the non- linearity is of self-defocusing type. This model arises in Bose-Einstein condensates trapped in a 2D optical lattice 共where t is time兲 关18,39兴 as well as light propagation in a periodic Kerr medium under paraxial approximation共wheret is the distance of propagation兲. In certain optical materials 共such as photorefractive crystals兲, the nonlinearity is of a different 共saturable兲 type. But those different nonlinearities often give qualitatively similar results as the cubic nonlin- earities above in a periodic lattice关13,15,16,36兴.

In this paper we take the lattice potential as

V共x,y兲=V₀共sin²x+ sin²y兲, 共2.2兲 whose periods along thexandydirections are both equal to

␲. This square-lattice potential can be readily engineered in Bose-Einstein condensates 关8,18兴 and optics 关5,6兴. This po- tential is separable, which makes our theoretical analysis a little easier. Similar analysis can be repeated for other types of periodic potentials with minimal changes.

Gap solitons in Eq. 共2.1兲are sought in the form

U共x,y,t兲=u共x,y兲e^i␮t, 共2.3兲 where the amplitude function u共x,y兲is real valued and sat- isfies the following equation:

u_xx+u_yy−关F共x兲+F共y兲兴u−␮u+␴u³= 0. 共2.4兲 Here F共x兲=V₀sin²x, and␮is a propagation constant. Note that the propagation constant in the above definition is the same as that in most papers on the stability theory of solitary waves in NLS-type equations关29,33兴. This is convenient for the comparison between our stability results in this paper and those in previous papers. In our previous publications on 2D gap solitons 关10,13兴, the propagation constant was defined with the opposite sign of that above. This should be kept in mind when quoting the results of关10,13兴.

When the amplitude of the gap soliton is infinitesimal, the nonlinear term in Eq. 共2.4兲 drops out. Solutions of the re-

maining linear equation are Bloch modes, and the corre- sponding propagation constants form Bloch bands. Between Bloch bands, band gaps may appear if the lattice potential is strong enough共see关10兴for details兲. When the amplitude of a gap soliton is small but not infinitesimal, the propagation constant lies near an edge of a Bloch band 共inside a band gap兲, and the gap soliton is a slowly modulated packet of the Bloch wave at this band edge. In this paper, we consider the stability properties of low-amplitude 2D gap solitons near edges of Bloch bands. We restrict ourselves to gap solitons that are slowly modulated single-Bloch-wave packets. At some band edges where two linearly independent Bloch modes exist, nonlinear coupling between the two Bloch- wave packets could generate more complex gap solitons such as vortex-array solitons and dipole-array solitons 关10,20兴. Such solitons consisting of two Bloch-wave packets are excluded from discussion in this paper.

Below we determine single-Bloch-wave packet solitons near a band edge␮₀. This derivation is a special case of our more general derivation in 关10兴 共see also 关40兴兲, and is thus only summarized here. This summary and proper extension are needed for the stability analysis in the next section.

Since the potential V共x,y兲is separable, a Bloch wave at this band edge has the formp₁共x兲p₂共y兲, wherep_1,2共x兲are 1D Bloch waves at 1D band edges ␻_1,2with

p_n

⬙

−F共x兲p_n−␻_np_n= 0, n= 1,2, 共2.5兲 and␮₀=␻₁+␻₂. The first few 1D band-edge values ␻_n and profiles of the corresponding Bloch waves p_n共x兲 have been displayed in关10兴 共forV₀= 6兲. These 1D Bloch functions have either period ␲ or 2␲ depending on the band edge, and p_n共x+␲_兲=⫾p_n共x兲. Since the sinusoidal potential F共x兲 is symmetric, Bloch functions p_n共x兲 are either symmetric or antisymmetric, i.e., p_n共−x兲=⫾p_n共x兲. Gap solitons in the form of a low-amplitude and slowly modulated packet of the Bloch wavep₁共x兲p₂共y兲have the following asymptotic expan- sions:

u=⑀u₀+⑀²u₁+⑀³u₂+ ¯ , 共2.6兲 and

␮=␮₀+␶⑀², 共2.7兲

where⑀Ⰶ1 is a soliton amplitude parameter,

u₀=A共X,Y兲p₁共x兲p₂共y兲 共2.8兲 is the leading-order solution,A共X,Y兲is a real-valued slowly varying envelope function,

X=⑀_共x−x₀兲, Y=⑀_共y−y₀兲 共2.9兲 are slow spatial variables, and 共x₀,y0兲is the center position of the envelope function. In order to simplify notations, we define the following operator:

L₀= ⳵²

⳵_x²⁺

⳵²

⳵_y^{2−关F共x兲+F共y兲兴−}␮+␴u²共x,y兲, 共2.10兲 then the solitary wave equation共2.4兲becomes

L₀u共x,y兲= 0. 共2.11兲

Since functionu共x,y兲contains fast and slow variables共x,y兲 and共X,Y兲, in our multiscale asymptotic analysis below, it is necessary to separate derivatives to these fast and slow vari- ables in L₀, so thatL₀is rewritten as

L₀=M₀+⑀M₁+⑀²M₂+⑀²␴共u₀+⑀u₁+ ¯兲², 共2.12兲 where

M₀= ⳵²

⳵_x²⁺

⳵²

⳵_y^{2−关F共x兲+F共y兲兴−}␮0, 共2.13兲

M₁= 2

冉

^⳵x^⳵⳵²_X⁺

⳵²

⳵_y⳵_Y

冊

^{, 共2.14兲}

M₂= ⳵²

⳵_X²⁺

⳵²

⳵_Y²⁻␶. 共2.15兲

Here the partial derivatives toxandyinM₀andM₁are with respect to the fast variables x and y only. Substituting the expansions 共2.6兲 and共2.12兲 into the solitary wave equation 共2.11兲, we get the following equations for the solutionsu_nat various orders of⑀:

⑀¹: M₀u₀= 0, 共2.16兲

⑀²: M₀u₁= −M₁u₀, 共2.17兲

⑀³: M₀u₂= −M₁u₁−共M₂+␴u₀²兲u₀, 共2.18兲

⑀⁴: M₀u₃= −M₁u₂−共M₂+ 3␴u₀²兲u₁, 共2.19兲

⑀⁵: M₀u₄= −M₁u₃−共M₂+ 3␴u₀²兲u₂− 3␴u₀u₁². 共2.20兲 The first-order equation共2.16兲is satisfied automatically. The second-order equation 共2.17兲 satisfies the Fredholm condi- tion, i.e., its inhomogeneous periodic termM₁u₀共in fast vari- ables x and y兲 is orthogonal to the homogeneous periodic solution p₁共x兲p₂共y兲 over a period of functions p₁共x兲 and p₂共y兲.

冕

₀^2␲

冕

₀^{2␲p1共x兲p2共y兲M1u0dxdy= 0. 共2.21兲}

Here the upper limits are taken as 2␲rather than␲because Bloch functions p_1,2共x兲 may have period 2␲ rather than ␲ 共see above兲. Hence Eq. 共2.17兲 admits a periodic solution, which is found to be

u₁=⳵_A

⳵_X␯₁共x兲p₂共y兲+⳵_A

⳵_Y^p1共x兲␯₂共y兲, 共2.22兲

where␯n共x兲 is the periodic solution of the equation

␯_n

⬙

−F共x兲␯_n−␻_n␯_n= − 2p_n

⬘

共x兲, n= 1,2. 共2.23兲 To make ␯_n共x兲 unique, we require its symmetry to be the same as that ofp_n

⬘

共x兲, i.e., the opposite of that ofp_n共x兲. Here

we did not add tou₁the homogeneous solution of Eq.共2.17兲.

The reason is that this homogeneous solution of Eq.共2.17兲is of the formB共X,Y兲p₁共x兲p₂共y兲. If this term were added tou₁ in Eq.共2.22兲, then theu₂solution would contain terms with the same symmetry ofp₁共x兲p₂共y兲in fast共x,y兲variables, plus the termB_X␯_1共x兲p₂共y兲+B_Yp₁共x兲␯_2共y兲. By using the Fredholm condition of theu₃ equation共2.19兲and the symmetry prop- erties of individual terms in fast variables as well as the relation 共2.27兲, we would find that L

1

eB= 0, where operator L₁^e is defined in Eq.共3.21兲. Hence the Bfunction would be a linear combination ofA_XandA_Y in the view that function A satisfies Eq. 共2.24兲. Then the resulting term B共X,Y兲p₁共x兲p₂共y兲 could be lumped into the leading-order term u₀ by slightly shifting the center position of A共X,Y兲, and thus could be removed fromu₁. Now we insert the above expressions ofu₀andu₁into theu₂equation共2.18兲. In order for thisu₂equation to admit a bounded共periodic兲solution in fast variablesxandy, the Fredholm condition gives the en- velope equation forA共X,Y兲 as关10兴

D₁

⳵²_A

⳵_X^2+D2

⳵²_A

⳵_Y²⁺␶A−␴␣0A³= 0, 共2.24兲 where

D_n⬅

冏

¹2 d²␻ dk²

冏

_␻=␻

n

, n= 1,2, 共2.25兲

are the second-order dispersion coefficients at 1D band edges

␻_n, and

␣₀=

冕

₀^␲

冕

₀^{␲p14共x兲p24共y兲dxdy}

冕

₀^␲

冕

₀^{␲p12共x兲p22共y兲dxdy⬎0. 共2.26兲}

In this derivation, we have used the relation关10,25兴

冕

₀^␲关2␯n

^⬘

^{共x兲+pn共x兲兴pn共x兲dx= −Dn}

冕

₀^{␲pn2共x兲dx,}

共2.27兲 wheren= 1 , 2. In the integrals of Eqs.共2.26兲and共2.27兲, the upper limits are taken as␲ rather than 2␲ of关10兴since the integrands here are periodic functions with period ␲ even when the Bloch functions p_1,2共x兲 have period 2␲.

The envelope equation 共2.24兲 is the familiar 2D NLS equation with constant coefficients. For the existence of gap solitons, the envelope function A must be a solitary wave, which decays to zero as共X,Y兲 approaches infinity. This re- quires that

sgn共D_1,2兲= − sgn共␴_{兲= − sgn共}␶_{兲. 共2.28兲} Under these conditions, the envelope equation共2.24兲admits a sign-definite ground-state 2D solitary wave which is bell shaped 共also called Townes profile兲. It also admits other types of excited-state solutions such as vortices. In this pa- per, we only consider the ground-state solitary wave solution of the envelope equation 共2.24兲. This ground-state envelope

solution leads to the simplest gap-soliton solutions in the lattice system共2.4兲.

It must be pointed out that at some 2D band edges with two linearly independent Bloch modes, the two Bloch modes can resonate with each other. In such cases, single-Bloch- wave packet solitons cannot exist共see band edgeE in关10兴兲.

The envelope equation 共2.24兲 has constant coefficients.

Thus it seems to suggest that the center of the envelope func- tion Acan move about freely in the共x,y兲plane. This is not so, however. We have shown in关10兴that due to an additional constraint on the solution, the center of the envelope function can only be located at four locations of the 2D lattice,

共x₀,y0兲=共0,0兲,

冉

^0,␲₂

冊

^,

冉

^␲₂^,0

冊

^,

冉

^␲₂^,␲₂

冊

^{. 共2.29兲}

The first location共x₀,y₀兲=共0 , 0兲is at a lattice site关minimum of potential V共x,y兲兴, hence the corresponding gap soliton is called an on-site soliton. The other locations are between lattice sites and correspond to off-site solitons. Notice that the second and third locations are equivalent to each other due to the symmetry of the lattice, thus they will be treated as the same. For illustration purposes, these three types of gap solitons near the two edges of the first Bloch band under focusing and defocusing nonlinearities are displayed in the upper and lower rows of Fig. 1, respectively 共with V₀= 6兲.

Their propagation constants are marked by circles in Fig.2.

We see that in both cases, on-site solitons have a single in- tensity peak共maximum兲, while off-site solitons have two or four equal intensity peaks. The difference between focusing and defocusing nonlinearities is that under focusing nonlin- earity 共in the semi-infinite band gap兲, those intensity peaks are all in phase 共see the upper row of Fig.1兲, while under defocusing nonlinearity 共in the first band gap兲, the intensity

(A) (B) (C)

(D) (E) (F)

FIG. 1. 共Color online兲 Upper row: three gap solitons near the right edge of the first Bloch band 共with focusing nonlinearity兲, where all the intensity peaks of solitons are in phase with each other. Lower row: three gap solitons near the left edge of the first band共with defocusing nonlinearity兲, where adjacent intensity peaks of solitons are out of phase with each other. The propagation con- stants of these solitons are marked as red circles in the power curves of Fig. 2. Circles on the background here represent locations of lattice sites. Some circles in共A兲and 共D兲 are skipped to show the solitons better. 关共A兲 and 共D兲兴 On-site gap solitons. 关共B兲, 共C兲, 共E兲, and共F兲兴Off-site gap solitons.

peaks are out of phase between adjacent sites共see the lower row of Fig. 1兲.

The power curve of gap solitons near band edges will prove to be very important in our stability analysis of the next section. Indeed, for sign-definite solitary waves, the VK stability criterion links the linear stability directly to the sign of the slope of the power curve 关29,33兴. For sign-indefinite gap solitons near band edges, we will show in the next sec- tion that this link also exists, but with important modifica- tions depending on the sign of nonlinearity共i.e., focusing or defocusing兲. The power of the soliton共2.3兲is defined as

P共␮_兲=具u,u典, 共2.30兲 where the inner product is defined as

具f,g典=

冕

_−⬁^⬁

冕

_−⬁^{⬁ f共x,y兲g共x,y兲dxdy. 共2.31兲}

Here the complex conjugate off is not used since most vari- ables involved in our inner products are real valued. For low-amplitude gap solitons near band edges, their power curve can be calculated asymptotically. By combining Eqs.

共2.6兲,共2.8兲, and共2.22兲, the asymptotic expansion for the gap soliton is

u共x,y;X,Y兲=⑀A共X,Y兲p₁共x兲p₂共y兲

+⑀²

冋

^⳵⳵AX␯_1共x兲p₂共y兲+

⳵_A

⳵_Y^p1共x兲␯_2共y兲

册

^{+O共⑀3兲.}

共2.32兲 Substituting this expansion into Eq.共2.30兲, we get the power functionP共␮_兲 as

P共␮兲=P₀+⑀P₁+⑀²P₂+ ¯, 共2.33兲 where

P₀=⑀²

冕

_−⬁^+⬁

冕

_−⬁^{+⬁A2共X,Y兲p12共x兲p22共y兲dxdy, 共2.34兲}

P₁= 2⑀³

冕

_−⬁^+⬁

冕

_−⬁^{+⬁A共X,Y兲p1共x兲p2共y兲}

⫻

冋

^⳵⳵AX␯_1共x兲p₂共y兲+

⳵_A

⳵_Y^p1共x兲␯_2共y兲

册

^{dxdy, 共2.35兲}

and so on. The integrands of the above integrals are products between periodic Bloch functions in fast variables共x,y兲and localized envelope functions in slow variables共X,Y兲. Using the formula in the Appendix, we find that the expressions of P₀andP₁for ⑀Ⰶ1 are

P₀=G

冕

_−⬁^⬁

冕

_−⬁^{⬁ A2共X,Y兲dXdY, 共2.36兲}

P₁= 0, 共2.37兲

where

G= 1

␲²

冕

₀^␲

冕

₀^{␲p12共x兲p22共y兲dxdy⬎0 共2.38兲}

is the average value of the squared Bloch-wave function p₁²共x兲p₂²共y兲. The errors in these expressions are exponentially small in⑀, thus they do not affect the power series expansion 共2.33兲of the power functionP共␮兲. Notice that the integral in Eq.共2.36兲is the power of the envelope solutionAin the 2D constant-coefficient NLS equation 共2.24兲. By variable scal- ings, it is easy to find that

冕

_−⬁^⬁

冕

_−⬁^{⬁ A2共X,Y兲dXdY=C0}

^冑

␣^D₀^{1D2, 共2.39兲} where

C₀= 11.70 共2.40兲

is the power of the ground-state soliton in the 2D NLS equa- tion with unit dispersion and nonlinearity coefficients. Insert- ing this formula into Eq.共2.36兲, we get

P₀=C₀G

冑

D₁D₂

␣₀ ^{. 共2.41兲}

ThisP₀is the limit power value of gap solitons on the edge of a Bloch band, and it is finite rather than infinite. Since P₁= 0, then in view of ␮−␮0=O共⑀²兲 out of Eq. 共2.7兲, the power series expansion共2.33兲gives

P共␮_兲=P₀+O共兩␮−␮_0兩兲, 共2.42兲 which indicates that the power curve near a band edge is a linear function of ␮.

It is noted that in the above asymptotic calculations of the power curves, the calculation results for the expansion coef- ficients P_ndo not depend on whether the solitons are on site or off site. Indeed, the difference between on-site and off-site solitons is that the slow-variable functions 关such asA共X,Y兲兴 in the perturbative solutionsu_n共x,y,X,Y兲are centered at dif- ferent positions but have the same profiles, and these slow- variable functions are separated from the fast-variable func- tions. Thus when calculating the soliton’s power, by using the integral formula of the Appendix, we see that the coeffi- cients P_n would be the same for both on-site and off-site solitons. This means that near band edges, the power differ- ence between on-site and off-site solitons is exponentially small in the soliton amplitude⑀.

To compute the whole power curve of gap solitons both near and far away from band edges, the numerical methods need to be used. Here we use the modified squared-operator iteration method proposed in 关41兴, which can converge to any gap soliton efficiently. The power curves of on-site soli- tons关see Figs.1共a兲and1共d兲兴bifurcating from the two edges of the first Bloch band under focusing and defocusing non- linearities are displayed in Fig. 2 共withV₀= 6兲. We see that both power curves attain a minimum value inside the band gaps, and the shapes of these curves resemble the letter “v.”

In addition, these power curves increase as ␮ approaches band edges. These features of the power curves appear to be common for 2D gap solitons bifurcated from band edges共see 关10兴for further examples兲. Note that the powers of these gap

solitons are much less than the powerC₀= 11.70 of 2D soli- tons in Eq.共2.4兲without the lattice potential. This means that these 2D gap solitons will not suffer the critical collapse of lattice-free 2D Kerr solitons 关42兴. The insets in Fig.2 show that these power curves approach a straight line near the band edges, which agrees with the analytical results in Eq.

共2.42兲. As␮approaches the right edge of the first band, the numerical power curve approaches a limit valueP₀= 1.9649, which agrees with the analytical formula共2.41兲. Similarly, as

␮ approaches the left edge of the first band, the numerical power curve approaches a limit valueP₀= 1.9567, which also agrees with the analytical formula共2.41兲.

III. LINEAR-INSTABILITY EIGENVALUES OF 2D GAP SOLITONS NEAR BAND EDGES

In this section, we study the linear stability of low- amplitude 2D gap solitons near band edges. These solitons are not sign definite in general 共see Fig. 1兲, thus the VK stability criterion关29,33兴does not apply to them. In addition to a zero eigenvalue, which is induced by the phase invari- ance of Eq.共2.1兲, these solitons also have three pairs of real or purely imaginary 共nonzero兲 eigenvalues. The reason for this is that near band edges, gap solitons are governed by an envelope equation, which is the familiar 2D NLS equation with constant coefficients 关see Eqs. 共2.24兲 and 共3.29兲 and 关10兴兴. Solitons in this 2D envelope equation have a unique discrete eigenvalue, which is zero. Corresponding to this zero eigenvalue, there are four eigenfunctions. One is in- duced by the phase invariance of the envelope soliton, an- other two are induced by the translational invariance of the envelope soliton共along the two spatial dimensions兲, and the fourth one is induced by the variation of the envelope soliton with respect to its propagation constant. For gap solitons in the full model 共2.1兲, the phase-invariance eigenmode of the

envelope soliton persists, but the other three eigenmodes of the envelope soliton do not persist because the full model 共2.1兲 is not translation invariant or its gap solitons do not have constant power. These three eigenmodes then have to bifurcate out from the zero eigenvalue. The bifurcated eigen- values always appear as pairs of real or purely imaginary eigenvalues since the system is Hamiltonian. If the pair of eigenvalues bifurcating from the propagation-constant- variation eigenmode of the envelope soliton are unstable, they lead to width instabilities of gap solitons where the soliton width either steadily increases共the soliton decays兲or decreases and relaxes into a periodic bound state共the soliton pulsates兲 depending on the initial power of the perturbed soliton 关13,43兴. If the two pairs of eigenvalues bifurcating from the translation-invariance eigenmodes of the envelope soliton are unstable, they lead to drift instabilities where the center of the soliton drifts away from its original location under perturbations. In this section, we focus on the pair of width-instability-type eigenvalues and derive their analytical expression near band edges by perturbation methods. This derivation does not depend on whether the gap solitons are on site or off site, thus our analytical formula is valid for both on-site and off-site solitons. This formula shows that one of this pair of eigenvalues is unstable if the slope of the power curve at the band edge has the opposite sign of non- linearity␴, or equivalently, if the soliton’s power near a band edge is lower than the limit power value on the band edge.

Our computations of the power curves for a number of on- site and off-site gap-soliton branches in the model共2.4兲show that this instability condition is always met under both focus- ing and defocusing cubic nonlinearities共see Figs.2and3for instance兲, thus these on-site and off-site 2D gap solitons near band edges are linearly unstable due to width instabilities.

Extensions of this analysis to other spatial dimensions共such as 1D and 3D兲will be discussed at the end of this section.

To study the linear stability of gap solitons u共x,y兲, we perturb them in the normal-mode form as

U共x,y,t兲=e^i␮t兵u共x,y兲+关v共x,y兲−␭⁻¹w共x,y兲兴e^i␭t +关v*共x,y兲+␭^*−1w^*共x,y兲兴e^−i␭*t其, 共3.1兲 where v,wⰆ1 are normal-mode perturbations, and the su- perscript ^* represents complex conjugation. Here we intro- duced a factor ␭⁻¹ in front of w, which gives the correct scaling of the normal-mode eigenfunction associated with the variation of the gap soliton with respect to its propaga- tion constant for small eigenvalues ␭. Substituting the per- turbed solution 共3.1兲 into the original evolution equation 共2.1兲and neglecting higher-order terms in 共v,w兲, we obtain the standard eigenvalue problem

L₁v=w, L₀w=␭²v, 共3.2兲

whereL₀has been defined in Eq.共2.10兲, andL₁is defined as

L₁=

⳵²

⳵_x²⁺

⳵²

⳵_y^{2−关F共x兲+F共y兲兴−}␮+ 3␴u²共x,y兲. 共3.3兲 Note that if 共␭,v,w兲 is an eigenmode of Eq. 共3.2兲, then so are 共−␭,v,w兲,共␭^*,v*,w^*兲, and共−␭^*,v*,w^*兲 as well. Thus eigenvalues of Eq. 共3.2兲 always appear in pairs or qua-

−4

−5 −4.5 −3.5

1 2 4

3

µ

Power

semi−infinite gap 1st−gap

FIG. 2. 共Color online兲Power curves of on-site 2D gap solitons bifurcated from the first Bloch band under focusing and defocusing nonlinearities, respectively共V0= 6兲. The asterisks “^*” are the ana- lytical limit power values共2.41兲 at the band edges. The insets are enlargements of the power curves showing linear dependence near band edges. Soliton profiles at the right and left marked points共red circles兲 are displayed in the upper and lower rows of Fig. 1, respectively.

druples. In addition, we have the relation 共2.11兲 as well as the relation

L₁u_␮=u. 共3.4兲

This second relation is obtained by taking the partial deriva- tive of Eq.共2.11兲with respect to␮. Furthermore, by taking the inner products of Eqs.共3.2兲first equation withu_␶and the second equation withu, and recalling the self-adjoint prop- erties of operators共L₀,L₁兲as well as the relations共2.11兲and 共3.4兲, we see that for nonzero eigenvalues␭, functionsuand vare orthogonal to each other, andu_␮andw are orthogonal to each other, i.e.,

具u,v典= 0, 共3.5兲

and

具u_␮,w典= 0. 共3.6兲

In our analysis below, we need to separate the partial de- rivatives in L₁ into those with respect to the fast and slow variables as we have done for the operator L₀. In doing so we get

L₁=M₀+⑀M₁+⑀²M₂+ 3⑀²␴共u₀+⑀u₁+ ¯兲², 共3.7兲

which is the analog of Eq.共2.12兲forL₀.

Our objective of this section is to determine the linear- stability eigenvalues ␭ and eigenfunctions 共v,w兲 for low- amplitude 2D gap solitons near edges of Bloch bands. Before detailed calculations, we first lay out our plans. The eigen- values of these gap solitons are clearly small since the soli- ton’s amplitude is low. Then in view of Eqs.共2.11兲and共3.2兲, we see that w should be proportional to u to the leading order. Indeed, we will find thatw⬃⑀u关see Eq.共3.50兲兴. Cor- respondingly, v⬃⑀u_␮ in view of Eqs. 共3.2兲 and 共3.4兲. This eigenmode is associated with the soliton’s variation with re- spect to the propagation constant ␮. Indeed, we will show that this mode bifurcates from the eigenmode of the 2D en- velope soliton’s zero eigenvalue induced by this envelope soliton’s variation with respect to its propagation constant 关see Eq.共3.32兲兴. This mode leads to width instabilities of gap solitons when it is unstable, and to width oscillations of gap solitons when it is stable 关13,43兴. But w cannot be exactly equal to⑀u, becausew=⑀u cannot satisfy the orthogonality condition 共3.6兲 in view that the power curve’s slope is not zero near band edges共see Fig.2兲. Calculation of the higher- order correction to the leading term⑀u ofwis a key step in our analysis. Through systematic perturbative calculations, we will manage to show that the higher-order correction tow is given in Eq.共3.50兲, where the function␨is given by Eq.

共3.45兲 and is proportional to ␭². Then by inserting this w formula into the orthogonality relation 共3.6兲, the expression for the eigenvalue ␭² will be obtained. It is noted that one can also determine the higher-order correction to the eigen- functionvand insert it into the orthogonality relation 共3.5兲, which will produce the same expression for ␭². But the former approach is a little simpler and thus will be adopted.

Now we start to calculate the power series expansions for the eigenmodes of low-amplitude gap solitons, which exist near band edges. For this purpose, we expand these eigen- functions and the eigenvalue into the following power series of ⑀:

v=v₀+⑀^v1+⑀^2v2+ ¯, 共3.8兲 w=w₀+⑀w₁+⑀²w₂+ ¯, 共3.9兲

␭²=⑀␩₁+⑀²␩₂+ ¯. 共3.10兲 Inserting these expansions and those of L₀ and L₁ into the eigenvalue problem共3.2兲, atO共1兲 we get

M₀w₀= 0, 共3.11兲

M₀v₀=w₀. 共3.12兲

From Eq.共3.11兲, we see thatw₀=h₀共X,Y兲p₁共x兲p₂共y兲. Due to the Fredholm condition for Eq. 共3.12兲, one must have h₀共X,Y兲= 0, thusw₀= 0. Then from Eq.共3.12兲, we get

v₀=␾_共X,Y兲p₁共x兲p₂共y兲. 共3.13兲 AtO共⑀_{兲, we get}

M₀w₁=␩₁^v₀, 共3.14兲

M₀v₁=w₁−M₁v₀. 共3.15兲 Applying the Fredholm condition to Eq. 共3.14兲, we see that

␩₁= 0, hencew₁=h₁共X,Y兲p₁共x兲p₂共y兲. Applying the Fredholm condition to Eq.共3.15兲and noticing thatM₁v₀andp₁共x兲p2共y兲 have the opposite symmetry in fast共x,y兲variables and hence the integral of their product is zero over one period of p₁共x兲 andp₂共y兲, we find thath₁共X,Y兲= 0, thus w₁= 0. In this case, the solutionv₁is

v₁=⳵␾

⳵_X␯1共x兲p₂共y兲+⳵␾

⳵_Y^p1共x兲␯2共y兲. 共3.16兲 Here we did not add Eq. 共3.15兲’s homogeneous solution of the form h共X,Y兲p₁共x兲p₂共y兲 intov₁because it turns out from later calculations that such a homogeneous term must be zero, thus we decided to leave it out at this early stage in order to simplify the analysis.

AtO共⑀²_{兲, we get}

M₀w₂=␩₂^v₀, 共3.17兲

M₀v₂=w₂−M₁v₁−共M₂+ 3␴u₀²兲v₀. 共3.18兲 Applying the Fredholm condition to Eq. 共3.17兲, we get ␩₂

= 0. Thus

w₂=␺共X,Y兲p₁共x兲p₂共y兲. 共3.19兲 The Fredholm condition for Eq.共3.18兲is

冕

₀^2␲

冕

₀^{2␲关w2−M1v1−共M2+ 3␴u02兲v0兴}

⫻p₁共x兲p₂共y兲dxdy= 0. 共3.20兲 Utilizing the expressions 共3.13兲, 共3.16兲, and 共3.19兲 for

共v₀,v₁,w₂兲as well as the relation共2.27兲, the above Fredholm condition reduces to

L₁^e␾= −␺, 共3.21兲

where

L₁^e⬅D₁

⳵²

⳵_X^2+D2

⳵²

⳵_Y²⁺␶− 3␴␣0A². 共3.22兲 Next we proceed to higher orders of ⑀. We will see that the equations for the next few higher-order terms in w are decoupled from the higher-order terms in v, thus we will consider higher-order terms inw only. AtO共⑀³兲, we have

M₀w₃=␩₃^v₀−M₁w₂. 共3.23兲 Its Fredholm condition shows that ␩₃= 0 关since the integral of M₁w₂ multiplying p₁共x兲p₂共y兲 is zero over one period of p₁共x兲andp₂共y兲兴. In this case, the solutionw₃ is then

w₃=

⳵␺

⳵_X␯1共x兲p₂共y兲+

⳵␺

⳵_Y^p1共x兲␯2共y兲. 共3.24兲 Here we did not add Eq. 共3.23兲’s homogeneous solution of the form h共X,Y兲p₁共x兲p₂共y兲 into w₃ either because such a term turns out to be zero as in the v₁ case.

AtO共⑀⁴_{兲, we have}

M₀w₄=␩₄^v₀−M₁w₃−共M₂+␴u₀²兲w₂. 共3.25兲 Its Fredholm condition gives

冕

₀^2␲

冕

₀^{2␲关␩4v0−M1w3−共M2+␴u02兲w2兴}

⫻p₁共x兲p₂共y兲dxdy= 0. 共3.26兲 Utilizing the expressions 共3.13兲, 共3.19兲, and 共3.24兲 for 共v₀,w₂,w3兲 as well as the relation 共2.27兲, the above Fred- holm condition reduces to

L₀^e␺= −␩₄␾, 共3.27兲

where

L₀^e⬅D₁ ⳵²

⳵_X^2+D2

⳵²

⳵_X²⁺␶−␴␣₀A². 共3.28兲 Now it is important to recognize that the two equations 共3.21兲and共3.27兲form the eigenvalue problem for the linear stability of ground-state envelope solitons Q共X,Y,T兲

=A共X,Y兲e^i␶Tin the 2D envelope NLS equation关10兴 iQ_Z−D₁Q_XX−D₂Q_YY+␴␣0兩Q兩²Q= 0. 共3.29兲 The linear-stability spectrum of these ground-state solitons in this 2D constant-coefficient NLS equation is well known. It has a single discrete eigenvalue, which is zero. Thus ␩₄= 0.

Corresponding to this zero eigenvalue, there are three dis- crete eigenfunctions.

␾=A_X, ␺= 0, 共3.30兲

␾=A_Y, ␺= 0, 共3.31兲

and

␾=A_␶, ␺=A. 共3.32兲

The eigenfunctions 共3.30兲 and 共3.31兲 are induced by the translational invariance of the envelope soliton in the enve- lope equation 共3.29兲, while the eigenfunction 共3.32兲 is in- duced by the variation of this envelope soliton with respect to its propagation constant ␶. It is noted that this envelope soliton also has a phase-invariance-induced eigenfunction at zero eigenvalue. But this phase-induced eigenmode does not satisfy the eigenvalue equations共3.21兲and共3.27兲of the en- velope soliton, because these eigenvalue relations are ob- tained with the eigenfunction scalings as used in Eq. 共3.1兲, which is not appropriate for the phase-induced eigenmode whose eigenvalue is always zero for both the envelope soli- ton and any gap soliton. Of the above eigenmodes of enve- lope solitons, we should point out that the propagation- constant-variation-induced eigenfunction 共3.32兲 with eigenvalue zero only exists in two dimensions because 2D envelope solitons in Eq.共3.29兲have constant powers for all propagation constants. This mode does not exist in other spa- tial dimensions共such as 1D or 3D兲because powers of enve- lope solitons are not constant in such cases.

As we have mentioned earlier, the eigenvalues we will calculate in this section bifurcate from the eigenmode of the 2D envelope soliton’s zero eigenvalue induced by this enve- lope soliton’s variation with respect to its propagation con- stant, thus we will take Eq.共3.32兲for the following analysis.

In this case, by comparing the expressions共3.13兲,共3.16兲, and 共3.19兲, and 共3.24兲 for 共v₀,v₁,w2,w₃兲 with the expressions 共2.8兲and共2.22兲for共u₀,u1兲, we see that

v₀=u_0␶, v₁=u_1␶, 共3.33兲

w₀=w₁= 0, w₂=u₀, w₃=u₁. 共3.34兲 In addition,

␩₁=␩₂=␩₃=␩₄= 0. 共3.35兲 Next we continue to pursue higher-order terms in the power series expansions of the eigenfunction w and eigen- value␭². The functionw₄satisfies Eq.共3.25兲. Utilizing Eqs.

共2.18兲,共3.34兲, and 共3.35兲, we find that

w₄=u₂+␨共X,Y兲p₁共x兲p₂共y兲. 共3.36兲 Here the inclusion of the homogeneous solution

␨共X,Y兲p₁共x兲p₂共y兲 inw₄ is necessary, and in fact crucial, in our analysis. To obtain the equation for w₅, we expand the eigenvalue problem共3.2兲toO共⑀⁵_兲and get

M₀w₅=␩₅^v₀−M₁w₄−共M₂+␴u₀²兲w₃− 2␴u₀u₁w₂. 共3.37兲 Inserting the expressions 共3.13兲, 共3.34兲, and 共3.36兲 for 共v₀,w₂,w3,w₄兲 into the above equation and utilizing the Fredholm condition on the u₃ equation 共2.19兲, one can readily show that the Fredholm condition for Eq. 共3.37兲 is satisfied only when ␩₅= 0. In this case, by utilizing the u₃ equation共2.19兲, the solutionw₅can be found to be