Two integrable differential-difference equations derived from NLS-type equation
aa r X i v : . [ n li n . S I] A p r Two integrable differential-difference equations derivedfrom NLS-type equation
Zong-Wei Xu † ,Guo-Fu Yu † and Yik-Man Chiang ‡ , † Department of Mathematics, Shanghai Jiao Tong University,Shanghai 200240, P.R. China ‡ Department of Mathematics, Hong Kong University of Science and Technology,Clear Water Bay, Kowloon, Hong Kong, P. R. China
Abstract
Two integrable differential-difference equations are derived from a (2+1)-dimensional modified Heisen-berg ferromagnetic equation and a resonant nonlinear Schr¨oinger equation respectively. Multi-solitonsolutions of the resulted semi-discrete systems are given through Hirota’s bilinear method. Elastic andinelastic interaction behavior between two solitons are studied through the asymptotic analysis. Dynam-ics of two-soliton solutions are shown with graphs. keywords: modified Heisenberg ferromagnetic system; Resonant nonlinear Schr¨oinger equation; Inte-grable discretization; Soliton interactions.Mathematics Subject Classification (2000). 35Q53, 37K10, 35C05, 37K40
Recently, integrable discretizations of integrable equations have been of considerable and current interestin soliton theory. As Suris mentioned [1], various approaches to the problem of integrable discretizationare currently developed, among which the Hirota’s bilinear method is very powerful and effective. Discreteanalogues of almost all interesting soliton equations, the KdV, the Toda chain, the sine-Gordon, etc., canbe obtained by the Hirota method. The purpose of this paper is to consider integrable discrete analogues oftwo nonlinear Schr¨odinger (NLS)-type system by Hirota method.The one-dimensional classical continuum Heisenberg models with different magnetic interactions havebeen settled as one of the interesting and attractive classes of nonlinear dynamical equations exhibiting thecomplete integrability on many occasions. As is well known, Heisenberg first proposed in 1928 the followingdiscrete (isotropic) Heisenberg ferromagnetic (DHF) spin chain [2]˙ S n = S n × ( S n + + S n − ) , (1)where S n = (s n , s n , s n ) ∈ R with | S n | = 1 and the overdot represents the time derivative with respect to t .The DHF chain plays an important role in the theory of magnetism.A performance of the standard continuous limit procedure leads DHF model (1) to the integrable Heisen-berg ferromagnetic model S t = S × S xx , (2)which is an important equation in condensed matter physics [3]. NLS-type equations are extensively usedto describe nonlinear water waves in fluids, ion-acoustic waves in plasmas, nonlinear envelope pulses in thefibers. It is known that HF is gauge equivalent to NLS equation and DHF is gauge equivalent to a kind ofdiscrete NLS-like equation [4].Higher dimensional nonlinear evolution equations are proposed to describe certain nonlinear phenomena.Due to the dependence on the additional spatial variables in higher dimensional systems, richer solutionstructure might appear, such as dromions, lumps, breathers and loop solitons.1n extension of Eq. (2) is the (2 + 1)-dimensional integrable modified HF system [5–8], as follows, u t + u xy + uw = 0 , (3a) v t − v xy − vw = 0 , (3b) w x + ( uv ) y = 0 , (3c)which is associated with the (2 + 1)-dimensional NLS equationi q τ + q ξτ − q Z ( | q | q ) η dξ = 0 , (4)where η is a spatial variable y . System (3) can also be used to model the biological pattern formation inreaction-diffusion process [8, 9]. In Ref. [6], system (3) was investigated through the prolongation structureand Lax representation. In Refs. [7, 8], integrable property of system (3) was studied through the Painlev´eanalysis, and some localized coherent and periodic solutions were given by means of the multi-linear variableseparation approach. Multi-soliton solutions of system (3) was derived in [10] by means of the Hirota bilinearmethod, and the double Wronskian solutions was given therein. Similar as the counterpart between DHF(1) and continuous HF (2), it is natural to consider discrete version of the (2 + 1)-dimensional modified HFsystem (3).A new integrable version of the nonlinear Schr¨odinger (NLS) equation, called the resonant NLS (RNLS)equation, iU t + U xx + α | U | U = β | U | | U | xx U , (5)was recently proposed [11] to describle low-dimensional gravity (the Jackiw-Teitelboim model) and responseof a medium to the action of a quasimonochromatic wave. Here α is a nonlinear coefficient, and β denotesthe strength of electrostriction pressure or diffraction. The term | U | xx / | U | on the righthand side of Eq. (5)is so called the ”quantum potential”. Moreover, Eq. (5) can model propagation of one-dimensional longmagnetoacoustic waves in a cold collisionless plasma subject to a transverse magnetic field [12]. Note thatwhen β < i Φ ξ + Φ ττ + σ | Φ | Φ = 0 . (6)When β >
1, it is not reducible to the usual NLS equation but to a reaction-diffusion (RD) system [12–14].The Lax pair in the 2 × β > − dimensional modified HF equation. In section 3, we study RNLS equation, including multi-solitonsolutions, semi-discrete analogue and dynamic properties of solutions. Finally, a short conclusion is given insection 4. N -soliton solution to modified HF system Through the variable transformation u = GF , v = HF , w = 2(ln( F )) xy , (7)2q.(3) transforms into the following bilinear form( D t + D x D y ) G • F = 0 , (8a)( D t − D x D y ) H • F = 0 , (8b) D x F • F + GH = 0 . (8c)The Hirota bilinear differential operator D mx D kt is defined by [17] D mx D ny a • b ≡ (cid:18) ∂∂x − ∂∂x ′ (cid:19) m (cid:18) ∂∂y − ∂∂y ′ (cid:19) n a ( x, y ) b ( x ′ , y ′ ) | x ′ = x,y ′ = y ,m, n = 0 , , , · · · . One-, two- and three-soliton solutions of bilinear equation set (8) are given in [10]. Note that the bilinearequations (8) are in Schr¨odinger type, here we present a compact form of multi-soliton solutions to thesystem (8). The two-soliton is given as F = 1 + a (1 , ∗ ) exp( η + ξ ) + a (1 , ∗ ) exp( η + ξ )+ a (2 , ∗ ) exp( η + ξ ) + a (2 , ∗ ) exp( η + ξ )+ a (1 , , ∗ , ∗ ) exp( η + η + ξ + ξ ) , (9) G = exp( η ) + exp( η ) + a (1 , , ∗ ) exp( η + η + ξ )+ a (1 , , ∗ ) exp( η + η + ξ ) , (10) H = exp( ξ ) + exp( ξ ) + a (1 , ∗ , ∗ ) exp( η + ξ + ξ )+ a (2 , ∗ , ∗ ) exp( η + ξ + ξ ) . (11)with η i = k i x + l i y + ω i t + η i , w i = − k i l i , (12) ξ i = p i x + q i y + Ω i t + ξ i , Ω i = p i q i , (13)and the coefficients are defined by a ( i, j ) = − k i − k j ) , (14) a ( i, j ∗ ) = − k i + p j ) , (15) a ( i ∗ , j ∗ ) = − p i − p j ) , (16) a ( i , i , · · · , i n ) = Y ≤ l 1) + u n v n = 0 . (46)When we take the continuum limit ǫ → 0, (119)-(129) reduce to (77)-(78) and2(ln F ) xx + uv = 0 . (47)Differentiating Eq.(130) with respect to variable y , we get Eq. (3c). Thus we regard (119)-(129) as a semi-discrete version of the HF system (3). In the following discussion we take the interval ǫ = 1 for the sake ofsimplicity.Following the Hirota method, we expand G n , H n and F n in series with a small parameter δ as F n = 1 + δ F (2) n + δ F (4) n + · · · + δ k F (2 k ) n + · · · , (48) G n = δG (1) n + δ G (3) n + · · · + δ (2 k +1) G (2 k +1) n + · · · , (49) H n = δH (1) n + δ H (3) n + · · · + δ (2 k +1) H (2 k +1) n + · · · . (50)Substituting the expansion into the above bilinear Eqs. (37)-(39), we find that there are only odd orderterms of δ in the first two equations while only even order terms appear in the third one. By the standarddirect perturbation method, we obtain the one-soliton solution G n = γ exp( η ) , H n = γ ′ exp( η ′ ) , F n = 1 − γ γ ′ β β ′ ( β β ′ − exp( η + η ′ ) , (51)where η = p t + q y + ln( β ) n, η ′ = p ′ t + q ′ y + ln( β ′ ) n, and β , β ′ satisfy β = 2 q − p q + p , (52) β ′ = 2 q ′ + p ′ q ′ − p ′ , (53) α , γ and α ′ , γ ′ are arbitrary constants. The two-soliton solution is presented as G n = δ [exp( η ) + exp( η )] + δ [ a exp( η + η + η ′ ) + a exp( η + η + η ′ )] , (54) H n = δ [exp( η ′ ) + exp( η ′ )] + δ [ b exp( η ′ + η ′ + η ) + b exp( η ′ + η ′ + η )] , (55) F n = 1 − δ h γ γ ′ β β ′ exp( η + η ′ )2( β β ′ − + γ γ ′ β β ′ exp( η + η ′ )2( β β ′ − + γ γ ′ β β ′ exp( η + η ′ )2( β β ′ − + γ γ ′ β β ′ exp( η + η ′ )2( β β ′ − i + δ χ [exp( η + η + η ′ + η ′ )] (56)5here a = − γ γ γ ′ ( β − β ) β ′ β β ′ − ( β β ′ − , a = − γ γ γ ′ ( β − β ) β ′ β β ′ − ( β β ′ − ,b = − γ γ ′ γ ′ ( β ′ − β ′ ) β β β ′ − ( β β ′ − , b = − γ γ ′ γ ′ ( β ′ − β ′ ) β β β ′ − ( β β ′ − ,χ = ( β − β ) ( β ′ − β ′ ) γ γ γ ′ γ ′ β β β ′ β ′ β β ′ − ( β β ′ − ( β β ′ − ( β β ′ − . (57)We can use the following compact expression for the above two-soliton solution, F n = 1 + a (1 , ∗ ) γ γ ′ γ ′ exp( η + η ′ ) + a (1 , ∗ ) γ γ ′ exp( η + η ′ )+ a (2 , ∗ ) γ γ ′ exp( η + η ′ ) + a (2 , ∗ ) γ γ ′ exp( η + η ′ )+ a (1 , , ∗ , ∗ ) γ γ ′ γ γ ′ exp( η + η + η ′ + η ′ ) , (58) G n = γ exp( η ) + γ exp( η ) + a (1 , , ∗ ) γ γ γ ′ exp( η + η + η ′ )+ a (1 , , ∗ ) γ γ γ ′ exp( η + η + η ′ ) , (59) H n = γ ′ exp( η ′ ) + γ ′ exp( η ′ ) + a (1 , ∗ , ∗ ) γ γ ′ γ ′ exp( η + η ′ + η ′ )+ a (2 , ∗ , ∗ ) γ γ ′ γ ′ exp( η + η ′ + η ′ ) , (60)where the coefficients are defined as a ( i, j ) = − β i − β j ) β i β j , (61) a ( i, j ∗ ) = − β i β ′ j ( β i β ′ j − , (62) a ( i ∗ , j ∗ ) = − β ′ i − β ′ j ) β ′ i β ′ j , (63)and a ( i, j, k ∗ ) , a ( i, j ∗ , k ∗ ) , a ( i, j, k ∗ , l ∗ ) satisfy the operation rule (17). In the same way, we can construct thethree-soliton solution as that in [20]. The above expressions of the one- and two-soliton solutions suggestthe exact N -soliton solution of Eqs. (37)-(39) in the following form F n = ( e ) X µ =0 , exp N X j =1 µ j η j + N X j = N +1 µ j η ′ j − N + N X ≤ k 0, Eqs. (119)-(120) tend to Eqs. (79)-(80). In the sequel weuse this discrete spacial step δ = 1. Multiplying (119) by 2(1 − a )( e v n +1 − v n + e v n − − v n ), and (120) by2(1 + a )( e u n +1 − u n + e u n − − u n ), adding and subtracting each other, respectively, yield2(1 + a ) h u n,t + a ( e u n +1 − u n + e u n − − u n − i ( e v n +1 − v n + e v n − − v n )+ 2(1 − a ) h v n,t − a ( e v n +1 − v n + e v n − − v n + 2) i ( e u n +1 − u n + e u n − − u n ) − α e u n + v n ( e u n +1 − u n + e u n − − u n )( e v n +1 − v n + e v n − − v n ) = 0 , (121)2(1 − a ) h u n,t + a ( e u n +1 − u n + e u n − − u n − i ( e v n +1 − v n + e v n − − v n )+ 2(1 + a ) h v n,t − a ( e v n +1 − v n + e v n − − v n + 2) i ( e u n +1 − u n + e u n − − u n )+ α e u n + v n ( e u n +1 − u n + e u n − − u n )( e v n +1 − v n + e v n − − v n ) = 0 . (122)Adding and subtracting (121)-(122) give us h u n,t + a ( e u n +1 − u n + e u n − − u n − i ( e v n +1 − v n + e v n − − v n )+ h v n,t − a ( e v n +1 − v n + e v n − − v n + 2) i ( e u n +1 − u n + e u n − − u n ) = 0 , (123)4 a h u n,t + a ( e u n +1 − u n + e u n − − u n − i ( e v n +1 − v n + e v n − − v n ) − a h v n,t − a ( e v n +1 − v n + e v n − − v n + 2) i ( e u n +1 − u n + e u n − − u n ) − αe u n + v n ( e u n +1 − u n + e u n − − u n )( e v n +1 − v n + e v n − − v n ) = 0 . (124)From (123) and (124), we get4( i − a ) h u n,t + a ( e u n +1 − u n + e u n − − u n − i ( e v n +1 − v n + e v n − − v n )+ 4( a + i ) h v n,t − a ( e v n +1 − v n + e v n − − v n + 2) i ( e u n +1 − u n + e u n − − u n )+ αe u n + v n ( e u n +1 − u n + e u n − − u n )( e v n +1 − v n + e v n − − v n ) = 0 . (125)Setting U n = exp (cid:16) u n + v n ia u n − v n (cid:17) , V n = U ∗ n = exp (cid:16) u n + v n − ia u n − v n (cid:17) , (126)or equivalently, u n = ln( U a − i a n V a + i a n ) , v n = ln( U a + i a n V a − i a n ) , (127)and substituting u n , v n by U n , V n into Eq. (125), we obtain a semi-discrete RNLS equation4( i − a ) h Re (cid:16) a − ia U n,t U n (cid:17) + a (cid:16)(cid:12)(cid:12)(cid:12)(cid:16) U n +1 U n (cid:17) a − i a (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:16) U n − U n (cid:17) a − i a (cid:12)(cid:12)(cid:12) − (cid:17)ih(cid:12)(cid:12)(cid:12)(cid:16) U n +1 U n (cid:17) a + i a (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:16) U n − U n (cid:17) a + i a (cid:12)(cid:12)(cid:12) i + 4( i + a ) h Re (cid:16) a + i a U n,t U n (cid:17) − a (cid:16)(cid:12)(cid:12)(cid:12)(cid:16) U n +1 U n (cid:17) a + i a (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:16) U n − U n (cid:17) a + i a (cid:12)(cid:12)(cid:12) + 2 (cid:17)ih(cid:12)(cid:12)(cid:12)(cid:16) U n +1 U n (cid:17) a − i a (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:16) U n − U n (cid:17) a − i a (cid:12)(cid:12)(cid:12) i + α (cid:12)(cid:12)(cid:12) U n (cid:12)(cid:12)(cid:12) (cid:16)(cid:12)(cid:12)(cid:12)(cid:16) U n +1 U n (cid:17) a + i a (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:16) U n − U n (cid:17) a + i a (cid:12)(cid:12)(cid:12) (cid:17)(cid:16)(cid:12)(cid:12)(cid:12)(cid:16) U n +1 U n (cid:17) a − i a (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:16) U n − U n (cid:17) a − i a (cid:12)(cid:12)(cid:12) (cid:17) = 0 . (128)In what follows we put a = 1 and α = − u n,t + (1 + e u n + v n )( e u n +1 − u n + e u n − − u n ) − , (129) v n,t − (1 + e u n + v n )( e v n +1 − v n + e v n − − v n ) + 2 = 0 . (130)13y the transformation e u n → u n and e v n → v n , Eqs. (129)-(130) can be expressed in an alternative simplerform u n,t − u n + (1 + u n v n )( u n +1 + u n − ) = 0 , (131) v n,t + 2 v n − (1 + u n v n )( v n +1 + v n − ) = 0 . (132) In this section, we construct the multi-soliton solutions of the semi-discrete RNLS equation (128) and henceshow its integrability. We rewrite Eqs. (116)-(118) with δ = a = 1 in following compact form (cid:16) D t − D n (cid:17) g n • f n = 0 , (133) (cid:16) D t + 2 − D n ) (cid:17) h n • f n = 0 , (134)2 sinh (cid:18) D n (cid:19) f n • f n − g n h n = 0 . (135)Similar to the continuous case, we expand f n , g n and h n into series with respect to a small parameter ǫ as follows f n = 1 + ǫ f (2) n + ǫ f (4) n + · · · + ǫ k f (2 k ) n + · · · , (136) g n = ǫg (1) n + ǫ g (3) n + · · · + ǫ k +1 g (2 k +1) n + · · · , (137) h n = ǫh (1) n + ǫ h (3) n + · · · + ǫ k +1 h (2 k +1) n + · · · . (138)We obtain the one-soliton solution g n = ǫ exp( η ) , h n = ǫ exp( η ′ ) , f n = 1 + ǫ ( β + β ′ η + η ′ ) , (139)where η = α t + β n + γ , η ′ = α ′ t + β ′ n + γ ′ , and β , β ′ satisfy the dispersion relation α + 4 sinh ( β , (140) α ′ − ( β ′ . (141)Here α , γ and α ′ , γ ′ are arbitrary parameters. The two-soliton solution is presented as follows g n = ǫ [exp( η ) + exp( η )] + ǫ [ ̺ exp( η + η + η ′ ) + ̺ exp( η + η + η ′ )] , (142) h n = ǫ [exp( η ′ ) + exp( η ′ )] + ǫ [ ς exp( η ′ + η ′ + η ) + ς exp( η ′ + η ′ + η )] , (143) f n = 1 + ǫ h 14 csch ( β + β ′ η + η ′ ) + 14 csch ( β + β ′ η + η ′ )+ 14 csch ( β + β ′ η + η ′ ) + 14 csch ( β + β ′ η + η ′ ) i + ǫ χ [exp( η + η + η ′ + η ′ )] , (144)where ̺ = ( e β + β ′ − e β + β ′ ) ( e β + β ′ − ( e β + β ′ − , ̺ = ( e β + β ′ − e β + β ′ ) ( e β + β ′ − ( e β + β ′ − ,ς = ( e β + β ′ − e β + β ′ ) ( e β + β ′ − ( e β + β ′ − , ς = ( e β + β ′ − e β + β ′ ) ( e β + β ′ − ( e β + β ′ − ,χ = ( e β ′ − e β ′ ) ( e β − e β ) e β + β + β ′ + β ′ ( e β + β ′ − ( e β + β ′ − ( e β + β ′ − ( e β + β ′ − . (145)14a) fisson (b) fusionFig. 3: The resonant interaction of two solitons of discrete RNLS. One soliton splits into two solitons, or,two solitons fuse into one.The coefficient χ plays a role of classification of soliton interactions. When χ is a finite number and notequal to zero, the regular soliton interaction exists. The resonant interactions of two solitons occur in Eq.(128) in the case of χ → ∞ or χ → 0. Our discussion is focused on the case of χ = 0 since the other casecan be analyzed in a similar way. Parametric conditions is given as follows( e β ′ − e β ′ )( e β − e β ) = 0 . (146)Choosing different factors of (146) as zero, we get two types of soliton resonances, i.e., the fission and fusion.Fig.3 (a) describes one soliton breaks into two solitons with the parameters selected as β = 0 . , β =1 . , β ′ = 0 . , β ′ = 0 . 3. Fig.3 (b) shows two solitons fuse into one soliton with the parameters β = 0 . , β =0 . , β ′ = 1 . , β ′ = 0 . f n = 1 + a (1 , ∗ ) exp( η + η ′ ) + a (1 , ∗ ) exp( η + η ′ )+ a (2 , ∗ ) exp( η + η ′ ) + a (2 , ∗ ) exp( η + η ′ )+ a (1 , , ∗ , ∗ ) exp( η + η + η ′ + η ′ ) , (147) g n = exp( η ) + exp( η ) + a (1 , , ∗ ) exp( η + η + η ′ )+ a (1 , , ∗ ) exp( η + η + η ′ ) ,h n = exp( η ′ ) + exp( η ′ ) + a (1 , ∗ , ∗ ) exp( η + η ′ + η ′ )+ a (2 , ∗ , ∗ ) exp( η + η ′ + η ′ ) , (148)where the coefficients are defined by a ( i, j ) = 4 sinh ( β i − β j , (149) a ( i, j ∗ ) = 14 csch ( β i + β ′ j , (150) a ( i ∗ , j ∗ ) = 4 sinh ( β ′ i − β ′ j . (151)15he exact N -soliton solution of eqs. 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