We show that a weak transverse spatial modulation in (2+1) nonlinear Schrödinger equation with saturable nonlinearity can result in nontrivial dynamics of radially symmetric solitons. In particular, in the case of hexagonal profile of the modulation the soliton moves chaotically

Radziunas, Mindaugas

Turaev, Dmitry

Vladimirov, Andrei G.

English

We show that a weak transverse spatial modulation in (2+1) nonlinear Schrödinger-type equation can result in nontrivial dynamics of a radially symmetric soliton. We provide examples of chaotic soliton motion in periodic media both for conservative and dissipative cases. We show that complex dynamics can persist even for soliton sizes greater than the modulation perio

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Chaotic soliton walk in periodically modulated media

Weierstraÿ-Institutfür Angewandte Analysis und Stohastikim Forshungsverbund Berlin e.V.Preprint ISSN 0946  8633Chaoti soliton walk in periodially modulatedmediaDmitry Turaev 1, Mindaugas Radziunas 2, Andrei Vladimirov 2submitted: 28th September 20071 Ben Gurion UniversityP.O.B. 65384105 Be'er ShevaIsraelE-mail: turaevmath.bgu,a.il 2 Weierstrass Institute for AppliedAnalysis and Stohastis,Mohrenstrasse 39,D - 10117 Berlin,GermanyE-Mail: radziunawias-berlin.devladimirwias-berlin.deNo. 1262Berlin 20072000 Mathematis Subjet Classiation. 78A60,37K45.Key words and phrases. nonlinear Shrödinger equation, soliton, haoti motion.1999 Physis and Astronomy Classiation Sheme. 42.65.Sf, 05.45.-a, 42.65.Tg.Edited byWeierstraÿ-Institut für Angewandte Analysis und Stohastik (WIAS)Mohrenstraÿe 3910117 BerlinGermanyFax: + 49 30 2044975E-Mail: preprintwias-berlin.deWorld Wide Web: http://www.wias-berlin.de/AbstratWe show that a weak transverse spatial modulation in (2+1) nonlinearShrödinger equation with saturable nonlinearity an result in nontrivial dy-namis of radially symmetri solitons. In partiular, in the ase of hexagonalprole of the modulation the soliton moves haotially.The nonlinear Shrödinger (NLS) equation plays a entral role in understanding var-ious physial phenomena in plasma physis, hydrodynamis, Bose-Einstein onden-sation, nonlinear optis. In partiular, NLS desribes pulse propagation in nonlinearbers and self-fousing of paraxial beams of light in a homogeneous Kerr medium[1℄. In the ase of purely ubi nonlinearity, (2+1)-dimensional NLS possesses aloalized solution, known as Townes mode [2℄. However, this solution is always un-stable: small perturbations lead to a ollapse, i.e. to an unbounded growth of theeld amplitude within a nite time interval. A suppression of the ollapse an beahieved by various means. In partiular, replaing the ubi nonlinearity with asaturable one ahieves an arrest of the ollapse and a stable self-ollimated propa-gation of a light beam [3℄. In a spatially homogeneous medium the paraxial beampropagates with a onstant veloity along a straight line. However, beause of re-ent developments in fabriation of mirostrutured wave-guiding materials knownas photoni rystals [4℄, there is a growing interest to the study of nonlinear beampropagation in various inhomogeneous settings [1, 5, 6, 7, 8℄.In our paper we study mobility properties of the stable solitons of (2+1)-dimensionalNLS equation with a saturable nonlinearity in the situation where the refrativeindex of the medium is modulated periodially in the transverse diretions. It isknown that the eetive partile approah is very eient in suh type of problems[9, 10, 11, 12, 13, 14, 15, 16, 17, 18℄. In partiular, in the ase of weak modulationamplitude we apply this approah to derive an equation, whih desribes the solitonas a Newtonian partile in the external potential reated by the refrative indexprole. This remains valid independently of the ratio of the soliton transverse sizeto the modulation period, even when the soliton is quite wide.We show that like a partile in a two-dimensional potential, the soliton in themedium with a transversely modulated refrative index an move both in a regularand haoti manner, and the hoie between these two types of motion is foremostdetermined by the geometry of the refrative index prole. Thus, when the refrativeindex forms a retangular lattie, the eetive potential is integrable, and the solitontransverse motion is very lose to integrable one for long time intervals. In this asethere are two typial dynamial regimes: the rst orresponds to low-energy quasi-periodi osillations around loal maximum of the refrative index (minimum of the1eetive potential), the seond orresponds to quasi-periodi osillations superim-posed on a onstant veloity drift. In the ase of hexagonal lattie the situation isdrastially dierent. Here, with the inrease of energy the osillations near a loalmaximum of refrative index beome haoti and transform into a random walk an unbounded transverse motion of the soliton wandering haotially between dif-ferent ells of the refrative index prole. Thus, our results show that even in simpleperiodi media a soliton an exhibit very ompliated motion patterns.Consider the equation∂tA = i∆A + Af(|A|2)+ iε2g (r)A, (1)where r = (x, y), ∆ = ∂xx + ∂yy, and A(r, t) is a omplex eld amplitude. Notethat when f is purely imaginary, Eq. (1) is Hamiltonian, with the energy funtionalgiven byH =12∫[|∂xA|2 + |∂yA|2 + Φ(|A|2)− ε2g(x, y)|A|2]dxdy,where Φ′ ≡ if . The onservation of energy H means that the purely imaginary forresponds to the light propagation in a transparent medium. In order to ensurethe stability of the soliton we use the saturable nonlinearity [3℄f(|A|2)=−i1 + |A|2 . (2)An important feature of Eq. (1) is that at ε = 0 it is invariant with respet to theGalilean transformation to a moving oordinate frame:A(r, t) → A (r − vt, t) exp(ir · v/2 − i|v|2t/4). (3)It follows that for ε = 0 any stationary solution of Eq. (1) oexists at with a family ofuniformly moving solutions parameterized by the veloity vetor v (here and belowwe use bold-fae letters to denote spatial 2-omponent vetors, while the entral dotdenotes a salar produt of suh vetors.The term iε2g in the right-hand side of Eq. (1) with small ε and real g (r) desribesthe spatial variation of the refrative index prole. If g is not spatially homogeneous,the translational and, hene, Galilean symmetries are broken at non-zero ε, whihresults in a non-trivial motion of the soliton in the (x, y)-plane.The propagation of paraxial light beams in a dissipative media an be desribedby the same equation (1), where the funtion f in the right-hand side is no longerpurely imaginary. In our simulations we take f real:f(|A|2)= −1 + G1 + |A|2 −Q1 + s|A|2 , (4)where G and Q are linear gain and, respetively, absorption oeients, and s > 1is the ratio of the saturation intensities of the gain and absorber media [19℄.2In this ase, the Hamiltonian struture of the equation is lost, while the Galileansymmetry is preserved at ε = 0. As we show below, the presene of this symmetryresults in a great similarity between the harater of soliton motion in the onserva-tive and dissipative ases, in spite of the dierene between the physial mehanismsof the soliton formation.Let Eq. (1) at ε = 0 have a radially symmetri stationary solitonA (r, t) = A0(r)eiω0t,where A0 → 0 exponentially fast as r → ∞ (we denote r = |r|). Sine the equationat ε = 0 is symmetri with respet to spatial translations, the vetor-funtionU = ∇A = rrA′0(r)satises LU = 0, where the operatorL : X 7→ [i (∆ − ω0) + f (E0) + E0f ′ (E0)] X + A20f ′ (E0) X∗yields the linearization of the right-hand side of Eq. (1) at the soliton solution. Herethe star denotes omplex onjugation, and E0 = |A0|2.Note that the Galilean symmetry of Eq. (1) implies the existene of the vetor-funtion Z suh that LZ = U. By dierentiating formula (3) with respet to v, itis easy to nd that Z = −irA0(r)/2.Let us dene the following inner produt of the funtions X and Y :〈X, Y 〉 =∫(XY + X∗Y ∗)dxdy.Aording to this denition, the adjoint to L operator L† reads asL† : X 7→ [i (∆ − ω0) + f (E0) + E0f ′ (E0)] X +[A20f′ (E0)]∗X∗.Like L, the operator L† has a non-trivial odd solution to L†U† = 0. Due to therotational symmetry we an write this solution in the form U† = rrU †(r), whereU †(r) is a salar funtion. An easy omputation gives〈Zx, U†x〉=〈Zy, U†y〉=∫Ψ(r)dxdy,where Ψ(r) = ∫ +∞r Im [U †(r′)A0(r′)] dr′, while the funtions Zx,y and Ux,y denotethe omponents of the vetor-funtions Z and U†.Below we assume that ∫ Ψ(r)dxdy 6= 0, i.e. 〈Z,U†〉 6= 0, whih means that thereis no solution to LX = Z. Note that in the Hamiltonian ase where f is purelyimaginary and A0 is real, it is easy to see that L† (iX) = iLX, whih implies thatU† = iU = irrA′0(r), and Ψ(r) = −12A20(r).In the non-Hamiltonian ase these relations are no longer true, and we do not haveexpliit formulas for U† and Ψ. 3At non-zero ε we will be looking for a slowly moving soliton solution in the form ofseries expansion{A0 [|r −R (εt)|] + εA1 [r − R (εt) , εt] + ε2A2 [r− R (εt) , εt] + . . .}eiω0t, (5)where R is the soliton enter position and A1,2,... desribe a small orretion to thesoliton shape. Substituting expansion (5) into Eq. (1) and olleting rst orderterms in ε we nd thatLA1 = −Ṙ · ∇A0 ≡ −Ṙ · U(the dot over R denotes the derivative with respet to the slow time εt). SineU = LZ, we an take A1 = −Ṙ · Z.Now, olleting the seond order terms in ε we obtainLA2 = −R̈ · Z(r − R) − ig(r)A0(|r −R|) −F(r −R), with F(−r) = F(r).Aording to the Fredholm alternative, the solvability of this equation with respetto A2 requires the orthogonality of its right-hand side to the solutions of the homo-geneous equation L†X = 0. So, by taking the inner produt of the right-hand sideto U †x(r − R) and U †y(r − R), and notiing that 〈F , U †x,y〉 = 0, we get the followingneessary solvability ondition:R̈∫Ψ(r)dxdy − 2∫g (r + R) Im[U†(r)A0(r)]dxdy = 0.Integrating by parts, we obtain nally the following equation for the soliton motion:R̈ = −∇V (R), (6)whereV (R) = −2∫g (r + R)Ψ(r)dxdy∫Ψ(r)dxdy. (7)As we mentioned, Ψ = −A20/2 in the Hamiltonian ase. Formula (7) generalizes theexpression for the eetive potential obtained for the integrable ase of 1D NLS inRefs. [11, 17℄.As we see, both in transparent and ative-dissipative media, the transverse solitonmotion is desribed, to the leading order, by the Hamiltonian equation (6). Thisis the equation of a unit mass partile moving in the external potential. Up to thefator of (−2) the potential is obtained by averaging the refrative index g with aweight determined by the soliton intensity. Note that Eqs. (6) and (7) are valid forarbitrary ratio of the soliton width to the harateristi period of the refrative indexmodulation. When this ratio is small, we obtain V (R) = −2g (R). As the ratiogrows, the averaging smooths the inhomogeneity of the refrative index. Therefore,when the soliton is suiently wide it moves, essentially, like a free partile.Being a Hamiltonian system with two degrees of freedom, equation (6) may exhibitboth regular and haoti dynamial regimes, depending on the shape of the potentialand on the value of energy. Below we show that the transverse dynamis of the4soliton depends strongly indeed on the struture of the refrative index prole. Weonsider square (g = g4) and hexagonal (g = g3) latties, dened bygj (r) = −l=j−1∑l=0cos(~kjl · ~r), ~ksl = k[cos(πl/j)sin(πl/j)].Here j = 4 and k = √2π/d for the square lattie, j = 3 and k = 4π/ (3d) for thehexagonal lattie, with d being the size of the lattie ell dened as the distanebetween a loal maximum of g and its nearest loal minimum. Aording to Eq. (7),the eetive potential that governs the soliton transverse motion is given byV (x, y) = −g(x, y)S (k) ,where the response oeient S (k) is dened byS (k) = 2∫cos (kx) Ψ(r)dxdy∫Ψ(r)dxdy.In the limit of a narrow soliton (k → 0), we have S (k) → 2, while in the oppositelimit k → ∞ the response funtion deays exponentially. In Fig. 1 we plot thegraph of amplitude prole and the response oeient S(k) for the solitons whosedynamis we studied in our numeri simulations. As it an be seen from the gure,S(k) is not negligibly small for the soliton sizes up to roughly twie the size of thelattie ell.r0 5 10 15|A0|0.00.51.01.52.02.53.0π/k0 5 10 15S(k)-0.50.00.51.01.52.0(a) (b)Figure 1: Considered soliton amplitude proles (a) and the response funtion S(k)(b). Solid and dotted lines orrespond, respetively, to onservative nonlinearity (2)and dissipative nonlinearity (4) with G = 2.11, Q = 2.0, and s = 10.5xycLxdLyaL bL Figure 2: Regular andhaoti soliton motion(blak and red urves)in the refrative indexprole forming square(a,b) and hexagonal(,d) lattie. Blue (red)olor indiates higher(lower) values of the re-frative index. The lefttwo gures (a,) orre-spond to onservativenonlinearity (2) and theright two gures (b,d) to dissipative nonlin-earity (4) with the pa-rameter values given inthe aption of Fig. 1.In the ase of square lattie, the eetive potentialV = S(k) [cos (kx) + cos (ky)]is separable, and therefore Eq. (6) is integrable, whih means a quasiperiodi motionfor the soliton. This result is onrmed by diret numerial integration of Eq. (1).Indeed, as we see in Figs. 2a,b, the soliton in the square lattie is either trappedin a lattie ell and osillates quasiperiodially in it, or the quasiperiodi osilla-tions aompany a onstant veloity drift. This piture is the same both for theonservative nonlinearity (2) and for the ative-dissipative ase (4). However, in thetime timesoliton     peak     coordinatea1)a2)c1)c2) d2)d1)b2)b1) Figure 3: Time dependenies ofthe oordinates x (blak) andy (red) of the soliton movingin square and hexagonal latties.The gures (a1) and (a2), (b1)and (b2), (1) and (2), (d1)and (d2) present the timetraesorresponding to the trajetoriesshown in gures 2a, 2b, 2, 2d,respetively.6xyeLxfLycL dLyaL bLFigure 4: Soliton motion in hexagonal refrative index proles with dierent sized = 8π/(3n) of an elementary ell. Here, n = 1 (panel a), n = 2 (b), n = 3 ()n = 4 (d), and n = 5 (e). The soliton size is approximately visible in panel f).7xyaLtimesolitoncoordinates cLsoitoncoordinates bLFigure 5: Soliton motion in a medium with nonzero spatial spetral ltering oe-ient δ = 0.01. Other parameter values are the same as in Figs. 2d and 3d1,d2.non-Hamiltonian ase we may see (Fig. 3b) a slow deay in the osillation amplitude,due to higher order orretions whih we negleted in our derivation of Eq. (6).The hexagonal refrative index lattie indues a dierent type of soliton motion, asthe hexagonal potential is known to reate haoti dynamis [20℄. Namely, while low-energy osillations near the minimum of the potential remain typially quasiperiodi(by KAM-theorem), the inrease of the energy leads to a random walk between theells. The numerial simulations both for onservative (2) and ative-dissipative(4) nonlinearities in Eq. (1) onrms this onlusion, showing either low-amplituderegular osillations or haoti wanderings of the soliton enter, see Figs. 2,d and3,d.The eet of the derease of the lattie ell size d on the soliton motion is illustratedin Fig. 4. For soliton diameters up to ∼ 4d we see a behavior resembling that of apartile in the potential. In Fig. 5 we present the results of numerial integration ofEqs. (1), (4) with the term δ∆A added into the right-hand side. This term with realand small δ > 0 orresponds to a spatial spetral ltering, whih breaks the Galileansymmetry of Eq. (1) at ε = 0. One an show that taking it into aount results inadding the dissipation term γṘ in the left-hand side of Eq. (6), with γ = O(δ/ε).Indeed, as we see in Fig. 5, we have at δ 6= 0 a steady derease of the amplitudeof osillations, regular and haoti alike. In this ase the soliton transverse motionnally halts at some position orresponding to a loal maximum of the refrativeindex.To onlude: we have established a haoti harater of motion of a soliton onhexagonal lattie engages itself in haoti motion, even for a weak amplitude of thespatial modulation. The eet ours both in the onservative ase and in the ase8of dissipative solitons.Referenes[1℄ Y.S. Kivshar, G.P. Agrawal, Optial solitons: from a ber to photoni rystals,Aademi Press, 2003.[2℄ R.Y. Chiao, E. Garmire, C.H. Townes, Phys. Rev. Lett., 13, p. 479, 1964.[3℄ M.G. Vakhitov, A.A. Kolokolov, Radiophys. Quantum Eletron, 16, p. 783,1975.[4℄ Photoni Crystals : Advanes in Design, Fabriation, and Charaterization, K.Bush, R.B. Wehrspohn, S. Lölkes, and H. Föll eds., Wiley-VCH, 2004.[5℄ F. Abdullaev, Theory of Solitons in Inhomogeneous Media, John Wiley & Sons,1994.[6℄ Nonlinear Photoni Crystals, Springer Series in Photonis, vol. 10, R.E. Slusherand B.J. Eggleton eds., Springer, 2003.[7℄ Y. 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Chaotic soliton walk in periodically modulated media

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