Commun. Theor. Phys. (Beijing China) 45 (2006) pp. 057 062 c International Academic Publishers Vol. 45 No. 6 June 5 2006 Jacobian Elliptic Function Method Solitary Wave Solutions for Hybrid Lattice Equation WANG Rui-Min DAI Chao-Qing 2 ZHANG Jie-Fang 2 College of Jinhua Professional Technology Jinhua 32000 China 2 Institute of Nonlinear Physics Zhejiang Normal University Jinhua 32004 China (Received October 2005; Revised December 6 2005) Abstract In this paper we have successfully extended the Jacobian elliptic function expansion approach to nonlinear differential-difference equations. The Hybrid lattice equation is chosen to illustrate this approach. As a consequence twelve families of Jacobian elliptic function solutions with different parameters of the Hybrid lattice equation are obtained. When the modulus m or 0 doubly-periodic solutions degenerate to solitonic solutions trigonometric function solutions PACS numbers: 05.45.Yv 02.30.Jr 02.30.Ik Key words: extended Jacobian elliptic function expansion approach hybrid lattice equation Jacobian elliptic function solutions solitonic solutions trigonometric function solutions Introduction In recent years noticeable progress has been made in the study of discrete spatial solitons in nonlinear media. [ Such solitons are intrinsic highly localized modes of nonlinear lattice [2 that form when discrete diffraction is balanced by nonlinearity. Soliton dynamics in spatially nonlinear systems plays a crucial role in the modelling of many phenomena in different fields ranging from condensed matter biophysics to mechanical engineering e.g. atomic chains [34 (discrete lattices) with onsite cubic nonlinearities molecular crystals [5 biophysical systems [6 electrical lattices [7 recently in arrays of coupled nonlinear optical wave guides. [89 Unlike difference equations which are fully discretized differentialdifference equations (DDES) are semi-discretized with some (or all) of their spacial variables discretized while time usually kept continuous. A wealth of information about integrable DDES can be found in papers by Suris [0 3 is book [4 in progress. Suris others have found many spatially discrete nonlinear models such as the Ablowitz Ladik lattice [5 the celebrated Toda lattice [6 the (2+) dimensional Toda lattice [7 the relativistic Toda lattice [3 the Volterra lattice models [0 the discrete mkdv equation [8 the Hybrid lattice [9 etc. For a given nonlinear partial differential equation (NPDE) many powerful methods have been developed to obtain exact solutions including the inverse scattering [20 the Bäcklund transformation [2 the homogenous balance method [22 the multilinear variable separation approach [23 the tanh method [24 the Jacobian elliptic function method [25 etc. However to our knowledge less work has been done to investigate exact solutions of DDES. For example Qian Lou [26 have successfully extended the multilinear variable separation approach to a special differential-difference Toda equation. More recently D. Baldwin et al. [27 presented an algorithm to find exact travelling wave solutions of differentialdifference equations in terms of tanh function found kink-type solutions in many spatially discrete nonlinear models. In this paper based on the extended Jacobian elliptic function algorithm for nonlinear differential equations presented by Zhen-Ya Yan [28 we develop this method to the nonlinear differential-difference equations. Though the modifications are slight they are of importance. For illustration we apply this method to the Hybrid lattice equation derive some Jacobian doubly periodic solutions. Particularly when the modulous m or 0 some solitonic solutions are generated. Our paper is organized as follows. In Sec. 2 the detailed derivation of the proposed method will be given. In Sec. 3 the application of the proposed method to the Hybrid lattice equation is illustrated. The final section is a short summary discussion. 2 Extended Jacobian Elliptic Function Method for Nonlinear DDES Consider a system of M polynomial DDES (u n+p (x)... u n+pk (x)... u n+p (x)... u n+p k (x)... u (r) n+p (x)... u (r) n+p k (x)) = 0 () where the dependent variable u n has M components u in the continuous variable x has N components x i the discrete variable n has Q components n j the k shift vectors p i Z Q u (r) (x) denotes the collection of mixed derivative terms of order r. Corresponding author E-mail: jf zhang@zjnu.cn
058 WANG Rui-Min DAI Chao-Qing ZHANG Jie-Fang Vol. 45 The main steps of the extended Jacobian elliptic function method are outlined as follows. Step When we seek the travelling wave solutions of Eq. () the first step is to introduce the wave transformation u n+ps (x) = φ n+ps (ξ n ) ξ n = Q i= d in i + N j= c jx j + ζ for any s (s =... k) where the coefficients c c 2... c N d d 2... d Q the phase ζ are all constants. In this way equation () becomes with (φ n+p (ξ n )... φ n+pk (ξ n )... φ n+p (ξ n )... φ n+p k (ξ n )... φ (r) n+p (ξ n )... φ (r) n+p k (ξ n )) = 0. (2) Step 2 We propose the following series expansion as a solution of Eq. () or (2): l φ n (ξ n ) = a 0 + f j ni (ξ n)[a j f ni (ξ n ) + b j g ni (ξ n ) i =... 8 (3) j= f n (ξ n ) = sn ξ n g n (ξ n ) = cn ξ n f n2 (ξ n ) = sn ξ n g n2 (ξ n ) = dn ξ n f n3 (ξ n ) = ns ξ n g n3 (ξ n ) = cs ξ n f n4 (ξ n ) = ns ξ n g n4 (ξ n ) = ds ξ n f n5 (ξ n ) = sc ξ n g n5 (ξ n ) = nc ξ n f n6 (ξ n ) = sd ξ n g n6 (ξ n ) = nd ξ n f n7 (ξ n ) = cd ξ n g n7 (ξ n ) = nd ξ n f n8 (ξ n ) = sc ξ n g n8 (ξ n ) = dc ξ n (4) where l is the integer to be determined later while sn ξ n = sn (ξ n m) cn ξ n = cn (ξ n m) dn ξ n = dn (ξ n m) are the Jacobian elliptic sine function the Jacobian elliptic cosine function the Jacobian elliptic function of the third kind other Jacobian functions which are denoted by Glaisher s symbols are generated by these three kinds of functions namely [2930 ns ξ n = sn ξ n nc ξ n = cn ξ n nd ξ n = dn ξ n sc ξ n = cs ξ n = sn ξ n cn ξ n ds ξ n = = dn ξ n cd ξ n = = sn ξ n (5) sd ξ n sn ξ n dc ξ n dn ξ n which have the relations sn 2 ξ n + cn 2 ξ n = dn 2 ξ n = m 2 sn 2 ξ n ns 2 ξ n = m 2 + ds 2 ξ n sc 2 ξ n + = nc 2 ξ n m 2 sd 2 ξ n + = nd 2 ξ n dc 2 ξ n + = m 2 + ( m 2 )nc 2 ξ n (6) (sn ξ n ) = cn ξ n dn ξ n (cn ξ n ) = sn ξ n dn ξ n (dn ξ n ) = m 2 sn ξ n cn ξ n (ns ξ n ) = ns ξ n sc ξ n dn ξ n (cs ξ n ) = ns ξ n ds ξ n (ds ξ n ) = ns ξ n cs ξ n (dc ξ n ) = ( m 2 )sc ξ n nc ξ n (sc ξ n ) = nc ξ n dc ξ n (nc ξ n ) = sc ξ n dc ξ n (sd ξ n ) = cd ξ n nd ξ n (nd ξ n ) = m 2 sd ξ n cd ξ n (7) with the modulus m (0 < m < ). In addition we know that sn(ξ + ξ 2 ) = ns(ξ + ξ 2 ) = sn ξ cn ξ 2 dn ξ 2 + sn ξ 2 cn ξ dn ξ m 2 sn 2 ξ sn 2 (8) ξ 2 Furthermore we assume cn(ξ + ξ 2 ) = nc(ξ + ξ 2 ) = cn ξ cn ξ 2 sn ξ dn ξ sn ξ 2 dn ξ 2 m 2 sn 2 ξ sn 2 ξ 2 (9) dn(ξ + ξ 2 ) = nd(ξ + ξ 2 ) = dn ξ dn ξ 2 m 2 sn ξ cn ξ sn ξ 2 cn ξ 2 m 2 sn 2 ξ sn 2 ξ 2 (0) cs(ξ + ξ 2 ) = sc(ξ + ξ 2 ) = cn ξ cn ξ 2 sn ξ dn ξ sn ξ 2 dn ξ 2 sn ξ cn ξ 2 dn ξ 2 + sn ξ 2 cn ξ dn ξ () ds(ξ + ξ 2 ) = sd(ξ + ξ 2 ) = dn ξ dn ξ 2 m 2 sn ξ cn ξ sn ξ 2 cn ξ 2 sn ξ cn ξ 2 dn ξ 2 + sn ξ 2 cn ξ dn ξ (2) cd(ξ + ξ 2 ) = dc(ξ + ξ 2 ) = cn ξ cn ξ 2 sn ξ dn ξ sn ξ 2 dn ξ 2 dn ξ dn ξ 2 m 2. sn ξ cn ξ sn ξ 2 cn ξ 2 (3) φ n+ps (ξ n ) = a 0 + l j= From the identities (8) (3) one obtains f j n+p s i (ξ n)[a j f n+ps i(ξ n ) + b j g n+ps i(ξ n ) i =... 8. (4) f n+ps (ξ n ) = sn ξ ncn(ϕ s )dn(ϕ s ) + sn(ϕ s )cn ξ n dn ξ n
No. 6 Jacobian Elliptic Function Method Solitary Wave Solutions for Hybrid Lattice Equation 059 with ϕ s satisfying g n+ps (ξ n ) = cn ξ ncn(ϕ s ) sn ξ n dn ξ n sn(ϕ s )dn(ϕ s ) f n+ps 2(ξ n ) = sn ξ ncn(ϕ s )dn(ϕ s ) + sn(ϕ s )cn ξ n dn ξ n g n+ps 2(ξ n ) = dn ξ ndn(ϕ s ) m 2 sn ξ n cn ξ n sn(ϕ s )cn(ϕ s ) f n+ps 3(ξ n ) = g n+ps 3(ξ n ) = cn ξ ncn(ϕ s ) sn ξ n dn ξ n sn(ϕ s )dn(ϕ s ) f n+ps 4(ξ n ) = g n+ps 4(ξ n ) = dn ξ ndn(ϕ s ) m 2 sn ξ n cn(ϕ s )sn(ϕ s )cn(ϕ s ) f n+ps 5(ξ n ) = sn ξ ncn(ϕ s )dn(ϕ s ) + sn(ϕ s )cn ξ n dn ξ n cn ξ n cn(ϕ s ) sn ξ n dn ξ n sn(ϕ s )dn(ϕ s ) g n+ps 5(ξ n ) = cn ξ n cn(ϕ s ) sn ξ n dn ξ n sn(ϕ s )dn(ϕ s ) f n+ps 6(ξ n ) = sn ξ ncn(ϕ s )dn(ϕ s ) + sn(ϕ s )cn ξ n dn ξ n dn ξ n dn(ϕ s ) m 2 sn ξ n cn ξ n sn(ϕ s )cn(ϕ s ) g n+ps 6(ξ n ) = dn ξ ndn(ϕ s ) m 2 sn ξ n cn ξ n sn(ϕ s )cn(ϕ s ) m 2 sn 2 ξ n sn 2 (ϕ s ) f n+ps 7(ξ n ) = cn ξ ncn(ϕ s ) sn ξ n dn ξ n sn(ϕ s )dn(ϕ s ) dn ξ n dn(ϕ s ) m 2 sn ξ n cn ξ n sn(ϕ s )cn(ϕ s ) g n+ps 7(ξ n ) = dn ξ ndn(ϕ s ) m 2 sn ξ n cn ξ n sn(ϕ s )cn(ϕ s ) f n+ps 8(ξ n ) = sn ξ ncn(ϕ s )dn(ϕ s ) + sn(ϕ s )cn ξ n dn ξ n cn ξ n cn(ϕ s ) sn ξ n dn ξ n sn(ϕ s )dn(ϕ s ) g n+ps 8(ξ n ) = dn ξ ndn(ϕ s ) m 2 sn ξ n cn ξ n sn(ϕ s )cn(ϕ s ) cn ξ n cn(ϕ s ) snξ n dn ξ n sn(ϕ s )dn(ϕ s ) (5) ϕ s = p s d + p s2 d 2 + + p sq d Q. (6) Meanwhile it is important to note that φ n+ps is a function of ξ n not ξ n+ps. Step 3 Determine the degree of the polynomial solutions (3) with ansatz (4). l is fixed by balancing the linear term of the highest order with the highest nonlinear term in Eq. (2). Suppose we are interested in balancing terms with shift p h then terms with shift other than p h say p s will not affect the balance since f n+ps i g n+ps i can be interpreted as being of degree zero in f n+ph i g n+ph i. Step 4 Substituting the ansatzs (3) (4) along with Eqs. (4) (5) into Eq. (2) then setting the coefficients of all powers like Jacobian elliptic functions to zero we will get a series of algebraic equations from which the constants a 0 a j b j (j = 2... l) c j (j = 2... N) are explicitly determined. Step 5 Substituting the obtained constants a 0 a j b j (j = 2... l) c j (j = 2... N) back into ansatz (3) we may get possible solutions of Eq. (). In addition we see that when m sn ξ cn ξ dn ξ degenerate as tanh ξ sech ξ sech ξ respectively while when m 0 sn ξ cn ξ dn ξ degenerate as sin ξ cos ξ Equations (3) (4) degenerate as solitonic solutions trigonometric function solutions of Eq. (). 3 Hybrid Lattice Equation In this section we apply the method developed in Sec. 2 to the Hybrid lattice equation obtain many types of solitonic solutions trigonometric function solutions. The Hybrid lattice equation [9 reads du n = ( + αu n + βu 2 dt n)(u n u n+ ) (7)
060 WANG Rui-Min DAI Chao-Qing ZHANG Jie-Fang Vol. 45 where u(n) u(n t) is a function of the discrete variable n the continuous variable t α β are two arbitrary parameters. In this case the variables u x n in Eq. () change into u x = t n = n respectively while p p 2 p 3 are replaced by p = p 2 = 0 p 3 = d = k c = c. We consider the transformations u n = φ n (ξ n ) ξ n = kn + ct + ζ (8) then equation (7) is reduced to u n+ = φ n+ (ξ n ) u n = φ n (ξ n ) (9) cφ n = ( + αφ n + βφ 2 n)(φ n φ n+ ). (20) We exp the solution of Eq. (20) in the form of Eqs. (3) (4). Balancing the linear term of the highest order with the highest nonlinear term in Eq. (20) we determined l =. 3. sn ξ n cn ξ n Solutions Substituting Eqs. (3) (4) with f n g n f n± g n± in Eqs. (4) (5) as well as l = that is φ n (ξ n ) = a 0 + a sn ξ n + b cn ξ n φ n+ (ξ n ) = a 0 + a [ sn ξn cn (k)dn (k) + sn (k)cn ξ n dn ξ n m 2 sn 2 ξ n sn 2 (k) φ n (ξ n ) = a 0 + a [ sn ξn cn (k)dn (k) sn (k)cn ξ n dn ξ n m 2 sn 2 ξ n sn 2 (k) [ cn ξn cn (k) sn ξ n dn ξ n sn (k)dn (k) + b m 2 sn 2 ξ n sn 2 (k) + b [ cn ξn cn (k) + sn ξ n dn ξ n sn (k)dn (k) m 2 sn 2 ξ n sn 2 (k) (2) (22) into Eq. (20) clearing the denominator setting the coefficients of all powers like dn ξ n sn i ξ n (i = 0 2 3) cn ξ n sn j ξ n (j = 0 2) to zero yields the following algebraic equations: 2a b (α + a 0 )sn(k) = 0 from which we have the following solutions: 2a 2 (a 0 + α)sn(k) 2b 2 (a 0 + α)sn(k)dn(k) = 0 a (a 2 b 2 )sn(k) ca m 2 sn(k) 4βa b 2 sn(k)dn(k) = 0 2a (βa 2 0 + αa 0 + βb 2 + )sn(k) + ca = 0 cb m 2 sn 2 (k) 4βb a 2 sn(k) + b (b 2 a 2 )sn(k)dn(k) = 0 2a b (α + a 0 )sn(k)dn(k) 2a b (α + a 0 )sn(k) = 0 2b (βa 2 0 + βb 2 + αa 0 + )sn(k)dn(k) + 4βb a 2 sn(k) cb = 0 (23) a 0 = α a = 0 b = ± m 4β α 2 sn(k) dn(k) c = ()sn(k) dn(k) (24) a 0 = α b = 0 a = ± m α 2 4β sn(k) c = ()sn(k) (25) where k is an arbitrary constant. According to Eqs. (8) (2) (24) when the parameters α β of Eq. (7) satisfy 4β α 2 > 0 we finally obtain u n = α ± m 4β α 2 sn(k) cn kn + ()sn(k) t + ζ. (26) dn(k) dn(k) According to Eqs. (8) (2) (25) when the parameters α β of Eq. (7) satisfy α 2 4β > 0 we finally obtain u 2n = α ± m α 2 4β sn(k) sn kn + ()sn(k) t + ζ. (27) 3.2 Other Jacobian Elliptic Function Solutions Similar to the procedure in Subsec. 3. we can also obtain other Jacobian elliptic function solutions by inserting Eqs. (3) (4) with f ni g ni f n±i g n±i (i = 2 8) in Eqs. (4) (5) as well as l =. These solutions has the forms: Case When the parameters α β satisfy 4β α 2 > 0 u 3n = α 4β ± α2 sn(k) dn kn + ()sn(k) cn(k) cn(k) u 4n = α (4β ± α2 )( m 2 ) sn(k) nd cn(k) t + ζ (28) kn + () sn(k) cn(k) t + ζ (29)
No. 6 Jacobian Elliptic Function Method Solitary Wave Solutions for Hybrid Lattice Equation 06 u 5n = α ± m (4β α 2 )( m 2 ) sn(k) dn(k) u 7n = α ± sn(k) sd Case 2 When the parameters α β satisfy α 2 4β > 0 u 6n = α ± sn(k) cs kn + ()sn(k) cn(k) cn(k) ns kn + ()sn(k) ds kn + ()sn(k) dn(k) u 8n = α ± sn(k) kn + ()sn(k) dn(k) dn(k) u 9n = α ± ()( m 2 ) sn(k) nc kn + ()sn(k) dn(k) dn(k) u 0n = α ± ()( m 2 ) sn(k) sc kn + ()sn(k) cn(k) cn(k) u n = α ± m α 2 4β sn(k) cd kn + ()sn(k) t + ζ u 2n = α ± sn(k) dc kn + ()sn(k) 3.3 Solitonic Solutions Trigonometric Function Solutions t + ζ. (30) t + ζ (3) t + ζ (32) t + ζ (33) t + ζ (34) t + ζ (35) (36) t + ζ. (37) When the modulus m 0 or the corresponding solitonic solutions trigonometric function solutions can be derived. (i) Solitonic solutions When m these doubly-periodic solutions (26) (27) degenerate as bell-kind kink-type solitons viz. u n = α ± 4β α 2 u 2n = α ± sinh(k)sech tanh(k)tanh kn + kn + sinh(k)t + ζ (38) tanh(k)t + ζ (39) These doubly-periodic solutions (3) (32) degenerate as singular solitonic solutions namely u 6n = α ± sinh(k)csch kn + sinh(k)t + ζ (40) u 7n = α ± tanh(k)coth kn + (ii) Trigonometric function solutions When m 0 these doubly-periodic solutions (3) (32) degenerate as u 6n = α ± u 7n = α ± tan(k)cot kn + These doubly-periodic solutions (34) (35) degenerate as u 9n = α ± sin(k)sec kn + u 0n = α ± tanh(k)t + ζ (4) tan(k)t + ζ (42) sin(k)csc kn + sin(k)t + ζ (43) sin(k)t + ζ (44) tan(k) tan kn + tan(k)t + ζ (45)
062 WANG Rui-Min DAI Chao-Qing ZHANG Jie-Fang Vol. 45 4 Summary Discussion In conclusion we have utilized the extended Jacobian elliptic function approach to construct twelve families of new doubly-periodic solutions for the Hybrid lattice equation. Difference of the parameters α β in the Hybrid lattice equation leads to different exact solutions. If the parameters satisfy α 2 4β > 0 the solutions (27) (3) (37) are obtained. If the parameters satisfy 4β α 2 > 0 the solutions (26) (28) (30) are derived. When the modulus m or 0 some of these obtained solutions degenerate as solitonic solutions trigonometric function solutions. In contrast to the derived solution in our present paper with some solutions based on partial differential equation we find that they are essentially equivalent. This method presented in this paper is only an initial work more work will be concerned. Although it is a pity that the combined solutions have not been found in the Hybrid lattice equation the abundant solutions obtained by this method imply that it is an excellent method to generalize to other differential-difference such as higher dimensional systems coupled nonlinear differential-difference equations. Therefore the more applications of this method to other nonlinear differential-difference systems deserve further investigation it may help us to find new interesting solutions to the given nonlinear discrete systems. In the future work we will also devote to searching for new method such as the generalized mapping method which is extensively applied in continuous partial differential equations to obtain new solutions of differential-difference equations. References [ D. Henning G.P. Tsiconis Phys. Rep. 307 (999) 333; O.M. Bcaun Y.S. Kivshav ibid. 306 (998) ; F. Ledever J.S. Aitchison Leshouches Workshop on Optical Solitons eds. V.E. Zakharov S. Wabnitz Springer-Verlag Berlin (999). [2 D.N. Christodoulides R.J. Joseph Opt. Lett. 93 (988) 794. [3 A.C. Scott L. Macheil Phys. Lett. A 98 (983) 87. [4 A.J. Sievers S. Takeno Phys. Rev. Lett. 6 (988) 970. [5 W.P. Su J.R. Schrieffer A.J. Heeger Phys. Rev. Lett. 42 (979) 698. [6 A.S. Davydov J. Theor. Biol. 38 (973) 599. [7 P. Marquii J.M. Bilbault M. Rernoissnet Phys. Rev. E 5 (995) 627. [8 H. Eisenberg Y. Silberberg R. Moraudotti A. Boyd J. Aitchison Phys. Rev. Lett. 8 (998) 3383. [9 R. Morotti U. Peschel J. Aitchison H. Eisenberg Y. Silberberg Phys. Rev. Lett. 83 (999) 2726. [0 Yu.B. Suris J. Phys. A: Math. Gen. 30 (997) 745. [ Yu.B. Suris J. Phys. A: Math. Gen. 30 (997) 2235. [2 Yu.B. Suris Rev. Math. Phys. (999) 727. [3 Yu.B. Suris Miura Transformations for Toda-type Integrable Systems with Applications to the Problem of Integrable Discretizations Sfb288 preprint 367 Department of Mathematics Technical University Berlin Berlin Germany (200). [4 Yu.B. Suris The Problem of Integrable Discretization: Hamiltonian Approach A skeleton of the book Sfb288 preprint 479 Department of Mathematics Technical University Berlin Berlin Germany (2002). [5 M.J. Ablowitz J.F. Ladik J. Math. Phys. 6 (975) 598; M.J. Ablowitz J.F. Ladik Stud. Appl. Math. 55 (976) 23. [6 E.G. Fan J. Phys. A: Math. Gen. 36 (2003) 7009. [7 K. Kajiwara J. Satsuma J. Math. Phys. 32 (99) 506. [8 M.J. Ablowitz J.F. Ladik Stud. Appl. Math. 57 (977). [9 R. Hirota M. Lwao Time-Discretization of Soliton Equation in: D. Levi O. Ragnisco eds. SIDE III- Symmetries Integrable of Difference Equations CRM Proc. Lect. Notes 25 Ams Providence Rhode Isl (2000) pp. 27 229. [20 C.S. Cardner J.M. Kruskal R.M. Miura Phys. Rev. Lett. 9 (967) 095. [2 H.D. Wahlquist F.B. Estabrook Phys. Lett. 3 (97) 386. [22 M.L. Wang Phys. Lett. A 99 (995) 69. [23 J.F. Zhang C.L. Zheng J.P. Meng J.P. Fang Chin. Phys. Lett. 20 (2003) 448. [24 E.G. Fan Phys. Lett. A 282 (200) 8. [25 E.G. Parkes B.R. Duffy P.C. Abbott Phys. Lett. A 295 (2002) 280. [26 X.M. Qian S.Y. Lou X.B. Hu J. Phys. A: Math. Gen. 37 (2003) 240. [27 D. Baldwin Ü. Göktas W. Hereman Comput. Phys. Commun. 62 (2004) 203. [28 Z.Y. Yan Comput. Phys. Commun. 48 (2002) 30. [29 D.V. Patrick Elliptic Function Elliptic Curves Cambridge University Press Cambridge (973). [30 K. Chamdrasekharan Elliptic Functions Springer- Verlag Berlin (985).