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A modified hyperbolic function method and its application to the Toda lattice
Applied Mathematics and Computation 172 (2006) 938–945 www.elsevier.com/locate/amc
A modified hyperbolic function method and its application to the Toda lattice Hongyan Zhi a,*, Xueqin Zhao a,b, Zhongyuan Yang c, Hongqing Zhang a
a
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China b Department of Mathematics, Qufu Normal University Qufu 273165, China c Department of Mathematics, University of Electronic Science and Technology of China, Chengdu 610054, China
Abstract In this paper, based on the computer symbolic system––Maple, we presented a modified hyperbolic function method to search for the exact solutions of nonlinear differential-difference equations (DDEs). The discrete Toda lattice is chosen to illustrate the method, and abundant new exact solutions are obtained, which include kink-shaped solitary wave solutions and other new types of exact solutions. This method can also be applied to many other nonlinear differential-difference equations (DDEs) in mathematics physics. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Toda lattice; Hyperbolic function expansion method; Kink-shaped solitary wave solution
*
Corresponding author. E-mail address: [email protected] (H. Zhi).
0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.02.036
H. Zhi et al. / Appl. Math. Comput. 172 (2006) 938–945
939
1. Introduction Differential-difference equations have played a crucial role in the modelling of many phenomena in different fields, which include condensed matter physics, mechanical engineering, vibrations in lattices, pulses in biological chains. DDEs also encounter such systems in numerical simulation of soliton dynamics in high energy physics where they arise as approximations of continuum models. A lot of work has been done on DDEs, including investigations of integrability criteria, the computation of densities, generalized and master symmetries, and recursion operators [1]. Notable is the work by Levi and colleagues [2,3], Yamilov and co-workers [4–11], where the classification of DDEs (into canonical forms), integrability tests, and connections between integrable PDEs and DDEs are analyzed in detail. Since the work of Fermi, Pasta and Ulam in the 1950s [12], the investigation of exact solutions of the DDEs has been the focus of many nonlinear studies (for references see [13,14]). Unlike difference equations which are fully discrete, DDEs are semi-discrete with some (or all) of their spacial variables discrete while time is usually kept continuous. So there are more difficulties in finding the exact solutions for DDEs. To our knowledge, little work is being done to symbolically compute exact solutions of DDEs. But there has been considerable work done on finding exact solutions to PDEs, various direct methods have become increasing attractive partly due to the availability of computer symbolic like Maple or Mathematica, such as homogeneous balance method [15], sine–cosine method [16], tanh method [17–19], the generalized Riccati equation method [20], the generalized projective Riccati equation method [21], Jacobin elliptic function method [22]. In particular, one of the most effective direct methods to construct exact solutions of PDEs is the hyperbolic function expansion method [23]. In this paper, we propose the modified hyperbolic function method for DDEs, abundant new solitary wave solutions of the Toda lattice are obtained.
2. Summary of the modified hyperbolic function expansion method (MHFEM) For a given nonlinear differential-difference equation ðkÞ ðkÞ D unþp1 ðxÞ; . . . ; unþpl ðxÞ; u0nþp1 ðxÞ; . . . ; u0nþpl ; unþp1 ðxÞ; . . . ; unþpl ðxÞ ¼ 0; ð1Þ (k)
where x = (x1, . . . , xm), n = (n1, . . . , ns), and m, s, p1, . . . , pl are integers, u (x) denotes the collection of mixed derivative terms of order k. We assume that any
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H. Zhi et al. / Appl. Math. Comput. 172 (2006) 938–945
arbitrary coefficients that parameterize the system are strictly positive and denoted by lower-case Greek letters. At first we introduce the following travelling wave transform s m X X d i ni þ kj xj þ d; ð2Þ un ¼ U n ðnn Þ; nn ¼ i¼1
j¼1
where di (i = 1, . . . , s), kj (j = 1, . . . , m), d are constants to be determined later. Substituting (2) into Eq. (1) yields an ordinary differential equation(ODE). ðkÞ ðkÞ H U nþp1 ; . . . ; U nþpl ; U 0nþp1 ; . . . ; U 0nþpl ; . . . ; U nþp1 ; . . . ; U nþpl ¼ 0. ð3Þ Step 1: We assume the solutions of Eq. (3) can be expressed in the form U n ðnn Þ ¼ a0 þ
n X
ai sinhi ðnn Þ þ
n X
i¼1
bi coshi ðnn Þ þ
i¼1
n X i¼1
i
ci þ gi sinh ðnn Þ . ðl1 sinhðnn Þ þ l2 coshðnn Þ þ l3 Þi
ð4Þ
From the properties of hyperbolic function, we have U nþr ðnnþr Þ ¼ a0 þ
n X ai ½sinhðnn ÞcoshðrdÞþ coshðnÞsinhðrdÞi i¼1
þbi ½coshðnn ÞcoshðrdÞþsinhðnn ÞsinhðrdÞi þ
n X i¼1
þ
n X i¼1
ci ½l1 sinhðnn ÞcoshðrdÞþ l1 coshðnn ÞsinhðrdÞþ l2 coshðnn ÞcoshðrdÞþ l2 sinhðnn ÞsinhðrdÞþ l3 i gi ½sinhðnn ÞcoshðrdÞþcoshðnn ÞsinhðrdÞi ; ½l1 sinhðnn ÞcoshðrdÞþ l1 coshðnn ÞsinhðrdÞþ l2 coshðnn ÞcoshðrdÞþ l2 sinhðnn ÞsinhðrdÞþ l3 i
ð5Þ where r is an arbitrary integer, a0, ai, bi, ci, gi, l1, l2, l3 are constants to be determined later, and n can be determined by homogeneous balance principle. Step 2: Substituting (4) and (5) into Eq. (3) and then setting all the coefficients of sinhi(nn)coshj(nn) (i = 0,1, j = 0, 1, . . .) to be zero, we obtain a set of algebraic equations with respect to a0, ai, bi, ci, gi, di, kj, l1, l2, l3. Step 3: With the help of Maple, we solve the over-determined nonlinear algebraic equations for a0, ai, bi, ci, gi, di, kj, l1, l2, l3. Step 4: Substituting the obtained conclusions in step 3 into Eq. (4), we can obtain the explicit and exact travelling solutions of Eq. (1).
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3. Solutions of discrete Toda lattice The discrete Toda lattice reads [24–26]: € un ðtÞ ¼ ðu_ n ðtÞ þ 1Þðun1 ðtÞ 2un ðtÞ þ unþ1 ðtÞÞ.
ð6Þ
The Toda lattices describe vibrations in mass-spring lattices with an exponential interaction force. And only one solitary wave solution was given in [26]. By the travelling transformation un ¼ U n ðnn Þ;
nn ¼ dn þ kt þ d;
ð7Þ
then Eq. (6) becomes k2 U 00n ðnn Þ ðkU 0n ðnn Þ þ 1ÞðU n1 ðnn1 Þ 2U n ðnn Þ þ U nþ1 ðnnþ1 ÞÞ ¼ 0.
ð8Þ
By the homogeneous balance principle, we can choose the solution of Eq. (8) in the form U n ¼ a0 þ a1 sinhðnn Þ þ a2 coshðnn Þ b1 b2 sinhðnn Þ þ þ ; l1 sinhðnn Þ þ l2 coshðnn Þ þ l3 l1 sinhðnn Þ þ l2 coshðnn Þ þ l3 ð9aÞ U nþ1 ¼ a0 þ a1 ½sinhðnn ÞcoshðdÞþ coshðnn ÞsinhðdÞ þ a2 ½coshðnn ÞcoshðdÞ þ sinhðnn ÞsinhðdÞ þ
b1 l1 sinhðnn ÞcoshðdÞ þ l1 coshðnn ÞsinhðdÞ þ l2 coshðnn ÞcoshðdÞþ l2 sinhðnn ÞsinhðdÞþ l3
þ
b2 ½sinhðnn ÞcoshðdÞ þ coshðnn ÞsinhðdÞ ; l1 sinhðnn ÞcoshðdÞ þ l1 coshðnn ÞsinhðdÞ þ l2 coshðnn ÞcoshðdÞþ l2 sinhðnn ÞsinhðdÞþ l3
ð9bÞ U n1 ¼ a0 þa1 ½sinhðnn ÞcoshðdÞ coshðnn ÞsinhðdÞ þa2 ½coshðnn ÞcoshðdÞ sinhðnn ÞsinhðdÞ þ
b1 l1 sinhðnn ÞcoshðdÞ l1 coshðnn ÞsinhðdÞ þl2 coshðnn ÞcoshðdÞ l2 sinhðnn ÞsinhðdÞ þl3
þ
b2 ½sinhðnn ÞcoshðdÞ coshðnn ÞsinhðdÞ : l1 sinhðnn ÞcoshðdÞ l1 coshðnn ÞsinhðdÞ þl2 coshðnn ÞcoshðdÞ l2 sinhðnn ÞsinhðdÞ þl3
ð9cÞ i
Substituting (9) into Eq. (8), collecting coefficients of sinh (nn)coshj(nn) (i = 0,1,j = 0, 1, 2 . . .), and setting them to be zero, we get a set of overdetermined algebraic equations with respect to a0, a1, a2, b1, b2, k, d, l1, l2, l3. By using of the Maple, solving the over-determined algebraic equations, we get the following results:
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Case 1 a2 ¼ b1 ¼ a1 ¼ l1 ¼ l3 ¼ 0; a0 ¼ a0 ; l2 ¼ b2 sinhðdÞ; k ¼ sinhðdÞ:
d ¼ d;
b2 ¼ b2 ;
Case 2 a1 ¼ a2 ¼ l1 ¼ b1 ¼ 0; b2 ¼ b2 ; a0 ¼ a0 ; 2 2b2 þ l22 d ¼ arccosh ; l2 ¼ l2 ; l3 ¼ l2 ; l22
k¼
2b2 : l2
Case 3 a1 ¼ a2 ¼ l1 ¼ 0; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 þ b22 l2 l3 ¼ ; b2 d ¼ arccosh
a0 ¼ a0 ;
2 2b2 þ l22 ; l22
b1 ¼ b1 ;
l2 ¼ l2 ;
k¼
2b2 ; l2
b2 ¼ b2 :
Case 4 a2 ¼ a1 ¼ l1 ¼ l3 ¼ 0; a0 ¼ a0 ; 2 k d ¼ arccosh þ 1 ; b2 ¼ b2 : 2
k ¼ k;
b1 ¼ ib2 ;
l2 ¼
Case 5 a1 ¼ l3 ¼ 0; l2 ¼ l2 ; a0 ¼ a0 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 l2 l21 l22 2b2 l b1 ¼ k ¼ 2 22 ; 2 2 l1 l2 l1 l2 k2 d ¼ arccosh 1 þ ; 2
a2 ¼ a1 ;
l1 ¼ l1 ;
b2 ¼ b2 :
Case 6 b1 ¼ l3 ¼ b1 ¼ 0; k ¼ 2i;
l2 ¼
l1 ¼ ib2 ; a21
a2 ¼ a1 ;
b2 ¼ b2 ;
ib2 ; a21
k2 d ¼ arccosh 1 þ : 2
a0 ¼ a0 ;
a1 ¼ a1 ;
2b2 ; k
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Case 7 b 2 ¼ b2 ; a1 ¼
l3 ¼ 0;
4ib2 ; 9b1
l1 ¼
a2 ¼ a1 ;
a0 ¼ a0 ;
b1 ¼ b1 ;
k ¼ 2i;
9ib21 ; 16b2
k2 d ¼ arccosh 1 þ : 2
9ib21 ; l2 ¼ 16b2 Case 8 a 2 ¼ a1 ; a1 ¼
l2 ¼ l1 ;
b2 ; 2l3
k ¼ k;
b1 ¼
a0 ¼ a 0 ;
b2 ¼ b2 ;
l3 ¼ l3 ;
l1 ¼ l1 ;
a0 ¼ a0 ;
a1 ¼ a1 ;
d ¼ d;
l3 b2 þ 2l21 a1 : 2l1
Case 9 a 2 ¼ a1 ; k ¼ k;
l2 ¼ l1 ; b2 ¼ l3 ¼ 0;
l1 ¼ l1 ; b1 ¼
d ¼ d;
l3 b2 þ 22l21 a1 : l1
Substituting Cases 1–9 into (9a), we can obtain the following exact solutions of Eq. (6). uðn; tÞ1 ¼ a0 þ tanhðnd þ sinhðdÞt þ dÞ; where a0, d, d are arbitrary constants. 2 2 2b þl 2 b2 sinh n arccosh 2l2 2 þ 2b t þ d l2 2 22 uðn; tÞ2 ¼ a0 þ ; 2b2 þl2 2b2 l2 cosh n arccosh l2 þ l 2 t þ d l2 2
where l2, b2, a0, d are arbitrary constants. b1 ffi pffiffiffiffiffiffiffiffi 2 2 2b2 þl2 b21 þb22 l2 2 l2 cosh n arccosh l2 t þ d þ þ 2b l2 b2 2 2 2 2b þl 2 b2 sinh n arccosh 2l2 2 þ 2b tþd l2 2 ffi ; þ pffiffiffiffiffiffiffiffi 2 2 2b þl b21 þb22 l2 2 l2 cosh n arccosh 2l2 2 þ 2b t þ d þ l2 b2
uðn; tÞ3 ¼ a0 þ
2
where l2, b2, b1, a0, d are arbitrary constants.
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2 ik k k þ 1 þ kt þ d þ uðn; tÞ4 ¼ a0 þ sech n arccosh 2 2 2 2 k tanh n arccosh þ 1 þ kt þ d ; 2 where k, a0, d are arbitrary constants. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 b2 l21 l22 uðn; tÞ5 ¼ a0 þ 2 ðl1 l22 Þl1 sinhðnn Þ þ l2 coshðnn Þ þ
b2 sinhðnn Þ ; l1 sinhðnn Þ þ l2 coshðnn Þ
2 l2 where nn ¼ n arccoshð1 þ 12 k2 Þ l2b2 l 2 t þ d and k, l1, l2, b2, a0, d are arbitrary 1 2 constants.
uðn; tÞ6 ¼ a0 þ a1 sinhðnn Þ þ a1 coshðnn Þ
ia21 sinhðnn Þ ; sinhðnn Þ coshðnn Þ
2
where nn ¼ n arccoshð1 þ k2 Þ þ 2it þ d and k, a1, d are arbitrary constants. uðn; tÞ7 ¼ a0
4ib2 sinhðnn Þ 4ib2 coshðnn Þ 16ib2 9b1 9b1 9b1 ðsinhðnn Þ coshðnn ÞÞ
16ib22 sinhðnn Þ ; 2 9b1 ðsinhðnn Þ coshðnn ÞÞ 2
where nn ¼ n arccoshð1 þ k2 Þ 2it þ d and k, b1, b2, a0, d are arbitrary constants. uðn; tÞ8 ¼ a0 þ
b2 sinhðnn Þ b2 coshðnn Þ b1 þ 2l3 2l3 l1 sinhðnn Þ l1 coshðnn Þ þ l3
b2 sinhðnn Þ ; l1 sinhðnn Þ l1 coshðnn Þ þ l3
where nn = nd + kt + d and k, l1, l3, d, b1, b2, a0, d are arbitrary constants. uðn; tÞ9 ¼ a0 þ a1 sinhðnn Þ þ a1 coshðnn Þ þ
b1 ; l1 sinhðnn Þ l1 coshðnn Þ
where nn = nd + kt + d and k, l1, d, b1, a1, a0, d are arbitrary constants. Remark. Among the above solutions, u(n, t)1, u(n, t)2, are kink-shaped solitary wave solutions. To our knowledge, all these solutions have not been given in literature.
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4. Conclusion In this paper, we presented the modified hyperbolic function method. The method is more powerful than the tanh-method proposed in [26]. The Toda lattice is chosen to illustrate the method such that nine solutions are obtained. Further work about various extension and improved hyperbolic function method needs us to find the more general ansa¨tzes.
References [1] W. Hereman et al., Symbolic computation of conserved densities, generalized symmetries, and recursion operators for nonlinear differential-difference equations, CRM Proc. and Lect. Notes, vol. 39, Amer. Math. Soc., Providence, RI, 2004. [2] D. Levi, O. Ragnisco, Lett. Nuovo Cimento 22 (1978) 691–696. [3] D. Levi, R.I. Yamilov, J. Math. Phys. 38 (1997) 6648–6674. [4] R.I. Yamilov, Classification of Toda type scalar lattices, in: V. Makhankov, I. Puzynin, O. Pashaev (Eds.), Proc. 8th Int. Workshop on Nonlinear Evolution Equations and Dynamical Systems, NEEDSÕ92, Dubna, USSR, 1992, World Scientific, Singapore, 1993, pp. 423–431. [5] R.I. Yamilov, J. Phys. A: Math. Gen. 27 (1994) 6839–6851. [6] V.E. Adler, S.I. Svinolupov, R.I. Yamilov, Phys. Lett. A 254 (1999) 24–36. [7] I.Yu. Cherdantsev, R.I. Yamilov, Physica D 87 (1995) 140–144. [8] I.Yu. Cherdantsev, R. Yamilov, Local master symmetries of differential-difference equations, in: D. Levi, L. Vinet, P. Winternitz (Eds.), Symmetries and Integrability of Difference Equations, CRM Proc. and Lect. Notes, vol. 9, Amer. Math. Soc., Providence, RI, 1996, pp. 51–61. [9] A.B. Shabat, R.I. Yamilov, Leningrad Math. J. 2 (1991) 377–400. [10] A.B. Shabat, R.I. Yamilov, Phys. Lett. A 227 (1997) 15–23. [11] R.I. Yamilov, Phys. Lett. A 160 (1991) 548–552. [12] E. Fermi, J. Pasta, S. Ulam, Collected Papers of Enrico Fermi II, University of Chicago Press, Chicago, IL, 1965, p. 978. [13] M. Hickman, W. Hereman, Proc. Roy. Soc. London A 459 (2003) 2705–2729. [14] G. Teschl, Jacobi operators and completely integrable nonlinear lattices, AMS Math. Surv. Monogr., vol. 72, Amer. Math. Soc., Providence, RI, 2000. [15] Z.Y. Yan, H.Q. Zhang, Appl. Math. Mech. 21 (2000) 382. [16] C. Yan, Phys Lett. A 224 (1996) 77. [17] S.A. Elwakil et al., Phys. Lett. A 299 (2002) 179. [18] E.G. Fan, Phys. Lett. A 277 (2000) 212. [19] Z.S. Lu¨, H.Q. Zhang, Phys. Lett. A 307 (2003) 269. [20] B. Li et al., Appl. Math. Comput. 152 (2004) 581. [21] R. Conte, M. Mustte, J. Phys. A 25 (1992) 5609. [22] H.T. Chen, H.Q. Zhang, Chaos Solitions and Fractals 20 (2004) 765; Y.Q. Yao, Chaos, Solitons and Fractals 23 (2005) 301. [23] Y.T. Gao, B. Tian, Comput. Phys. Commun. 133 (2001) 158; B. Tian, Y.T. Gao, Z. Naturscrift A 57 (2002) 39. [24] M. Toda, Theory of Nonlinear Lattices, Springer-Verlag, Berlin, 1981. [25] Yu.B. Suris, J. Phys. A: Math. Gen. 30 (1997) 1745. [26] D. Baldwin et al., Comput. Phys. Commun. 162 (2004) 203.
A modified hyperbolic function method and its application to the Toda lattice Hongyan Zhi a,*, Xueqin Zhao a,b, Zhongyuan Yang c, Hongqing Zhang a
a
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China b Department of Mathematics, Qufu Normal University Qufu 273165, China c Department of Mathematics, University of Electronic Science and Technology of China, Chengdu 610054, China
Abstract In this paper, based on the computer symbolic system––Maple, we presented a modified hyperbolic function method to search for the exact solutions of nonlinear differential-difference equations (DDEs). The discrete Toda lattice is chosen to illustrate the method, and abundant new exact solutions are obtained, which include kink-shaped solitary wave solutions and other new types of exact solutions. This method can also be applied to many other nonlinear differential-difference equations (DDEs) in mathematics physics. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Toda lattice; Hyperbolic function expansion method; Kink-shaped solitary wave solution
*
Corresponding author. E-mail address: [email protected] (H. Zhi).
0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.02.036
H. Zhi et al. / Appl. Math. Comput. 172 (2006) 938–945
939
1. Introduction Differential-difference equations have played a crucial role in the modelling of many phenomena in different fields, which include condensed matter physics, mechanical engineering, vibrations in lattices, pulses in biological chains. DDEs also encounter such systems in numerical simulation of soliton dynamics in high energy physics where they arise as approximations of continuum models. A lot of work has been done on DDEs, including investigations of integrability criteria, the computation of densities, generalized and master symmetries, and recursion operators [1]. Notable is the work by Levi and colleagues [2,3], Yamilov and co-workers [4–11], where the classification of DDEs (into canonical forms), integrability tests, and connections between integrable PDEs and DDEs are analyzed in detail. Since the work of Fermi, Pasta and Ulam in the 1950s [12], the investigation of exact solutions of the DDEs has been the focus of many nonlinear studies (for references see [13,14]). Unlike difference equations which are fully discrete, DDEs are semi-discrete with some (or all) of their spacial variables discrete while time is usually kept continuous. So there are more difficulties in finding the exact solutions for DDEs. To our knowledge, little work is being done to symbolically compute exact solutions of DDEs. But there has been considerable work done on finding exact solutions to PDEs, various direct methods have become increasing attractive partly due to the availability of computer symbolic like Maple or Mathematica, such as homogeneous balance method [15], sine–cosine method [16], tanh method [17–19], the generalized Riccati equation method [20], the generalized projective Riccati equation method [21], Jacobin elliptic function method [22]. In particular, one of the most effective direct methods to construct exact solutions of PDEs is the hyperbolic function expansion method [23]. In this paper, we propose the modified hyperbolic function method for DDEs, abundant new solitary wave solutions of the Toda lattice are obtained.
2. Summary of the modified hyperbolic function expansion method (MHFEM) For a given nonlinear differential-difference equation ðkÞ ðkÞ D unþp1 ðxÞ; . . . ; unþpl ðxÞ; u0nþp1 ðxÞ; . . . ; u0nþpl ; unþp1 ðxÞ; . . . ; unþpl ðxÞ ¼ 0; ð1Þ (k)
where x = (x1, . . . , xm), n = (n1, . . . , ns), and m, s, p1, . . . , pl are integers, u (x) denotes the collection of mixed derivative terms of order k. We assume that any
940
H. Zhi et al. / Appl. Math. Comput. 172 (2006) 938–945
arbitrary coefficients that parameterize the system are strictly positive and denoted by lower-case Greek letters. At first we introduce the following travelling wave transform s m X X d i ni þ kj xj þ d; ð2Þ un ¼ U n ðnn Þ; nn ¼ i¼1
j¼1
where di (i = 1, . . . , s), kj (j = 1, . . . , m), d are constants to be determined later. Substituting (2) into Eq. (1) yields an ordinary differential equation(ODE). ðkÞ ðkÞ H U nþp1 ; . . . ; U nþpl ; U 0nþp1 ; . . . ; U 0nþpl ; . . . ; U nþp1 ; . . . ; U nþpl ¼ 0. ð3Þ Step 1: We assume the solutions of Eq. (3) can be expressed in the form U n ðnn Þ ¼ a0 þ
n X
ai sinhi ðnn Þ þ
n X
i¼1
bi coshi ðnn Þ þ
i¼1
n X i¼1
i
ci þ gi sinh ðnn Þ . ðl1 sinhðnn Þ þ l2 coshðnn Þ þ l3 Þi
ð4Þ
From the properties of hyperbolic function, we have U nþr ðnnþr Þ ¼ a0 þ
n X ai ½sinhðnn ÞcoshðrdÞþ coshðnÞsinhðrdÞi i¼1
þbi ½coshðnn ÞcoshðrdÞþsinhðnn ÞsinhðrdÞi þ
n X i¼1
þ
n X i¼1
ci ½l1 sinhðnn ÞcoshðrdÞþ l1 coshðnn ÞsinhðrdÞþ l2 coshðnn ÞcoshðrdÞþ l2 sinhðnn ÞsinhðrdÞþ l3 i gi ½sinhðnn ÞcoshðrdÞþcoshðnn ÞsinhðrdÞi ; ½l1 sinhðnn ÞcoshðrdÞþ l1 coshðnn ÞsinhðrdÞþ l2 coshðnn ÞcoshðrdÞþ l2 sinhðnn ÞsinhðrdÞþ l3 i
ð5Þ where r is an arbitrary integer, a0, ai, bi, ci, gi, l1, l2, l3 are constants to be determined later, and n can be determined by homogeneous balance principle. Step 2: Substituting (4) and (5) into Eq. (3) and then setting all the coefficients of sinhi(nn)coshj(nn) (i = 0,1, j = 0, 1, . . .) to be zero, we obtain a set of algebraic equations with respect to a0, ai, bi, ci, gi, di, kj, l1, l2, l3. Step 3: With the help of Maple, we solve the over-determined nonlinear algebraic equations for a0, ai, bi, ci, gi, di, kj, l1, l2, l3. Step 4: Substituting the obtained conclusions in step 3 into Eq. (4), we can obtain the explicit and exact travelling solutions of Eq. (1).
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3. Solutions of discrete Toda lattice The discrete Toda lattice reads [24–26]: € un ðtÞ ¼ ðu_ n ðtÞ þ 1Þðun1 ðtÞ 2un ðtÞ þ unþ1 ðtÞÞ.
ð6Þ
The Toda lattices describe vibrations in mass-spring lattices with an exponential interaction force. And only one solitary wave solution was given in [26]. By the travelling transformation un ¼ U n ðnn Þ;
nn ¼ dn þ kt þ d;
ð7Þ
then Eq. (6) becomes k2 U 00n ðnn Þ ðkU 0n ðnn Þ þ 1ÞðU n1 ðnn1 Þ 2U n ðnn Þ þ U nþ1 ðnnþ1 ÞÞ ¼ 0.
ð8Þ
By the homogeneous balance principle, we can choose the solution of Eq. (8) in the form U n ¼ a0 þ a1 sinhðnn Þ þ a2 coshðnn Þ b1 b2 sinhðnn Þ þ þ ; l1 sinhðnn Þ þ l2 coshðnn Þ þ l3 l1 sinhðnn Þ þ l2 coshðnn Þ þ l3 ð9aÞ U nþ1 ¼ a0 þ a1 ½sinhðnn ÞcoshðdÞþ coshðnn ÞsinhðdÞ þ a2 ½coshðnn ÞcoshðdÞ þ sinhðnn ÞsinhðdÞ þ
b1 l1 sinhðnn ÞcoshðdÞ þ l1 coshðnn ÞsinhðdÞ þ l2 coshðnn ÞcoshðdÞþ l2 sinhðnn ÞsinhðdÞþ l3
þ
b2 ½sinhðnn ÞcoshðdÞ þ coshðnn ÞsinhðdÞ ; l1 sinhðnn ÞcoshðdÞ þ l1 coshðnn ÞsinhðdÞ þ l2 coshðnn ÞcoshðdÞþ l2 sinhðnn ÞsinhðdÞþ l3
ð9bÞ U n1 ¼ a0 þa1 ½sinhðnn ÞcoshðdÞ coshðnn ÞsinhðdÞ þa2 ½coshðnn ÞcoshðdÞ sinhðnn ÞsinhðdÞ þ
b1 l1 sinhðnn ÞcoshðdÞ l1 coshðnn ÞsinhðdÞ þl2 coshðnn ÞcoshðdÞ l2 sinhðnn ÞsinhðdÞ þl3
þ
b2 ½sinhðnn ÞcoshðdÞ coshðnn ÞsinhðdÞ : l1 sinhðnn ÞcoshðdÞ l1 coshðnn ÞsinhðdÞ þl2 coshðnn ÞcoshðdÞ l2 sinhðnn ÞsinhðdÞ þl3
ð9cÞ i
Substituting (9) into Eq. (8), collecting coefficients of sinh (nn)coshj(nn) (i = 0,1,j = 0, 1, 2 . . .), and setting them to be zero, we get a set of overdetermined algebraic equations with respect to a0, a1, a2, b1, b2, k, d, l1, l2, l3. By using of the Maple, solving the over-determined algebraic equations, we get the following results:
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Case 1 a2 ¼ b1 ¼ a1 ¼ l1 ¼ l3 ¼ 0; a0 ¼ a0 ; l2 ¼ b2 sinhðdÞ; k ¼ sinhðdÞ:
d ¼ d;
b2 ¼ b2 ;
Case 2 a1 ¼ a2 ¼ l1 ¼ b1 ¼ 0; b2 ¼ b2 ; a0 ¼ a0 ; 2 2b2 þ l22 d ¼ arccosh ; l2 ¼ l2 ; l3 ¼ l2 ; l22
k¼
2b2 : l2
Case 3 a1 ¼ a2 ¼ l1 ¼ 0; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 þ b22 l2 l3 ¼ ; b2 d ¼ arccosh
a0 ¼ a0 ;
2 2b2 þ l22 ; l22
b1 ¼ b1 ;
l2 ¼ l2 ;
k¼
2b2 ; l2
b2 ¼ b2 :
Case 4 a2 ¼ a1 ¼ l1 ¼ l3 ¼ 0; a0 ¼ a0 ; 2 k d ¼ arccosh þ 1 ; b2 ¼ b2 : 2
k ¼ k;
b1 ¼ ib2 ;
l2 ¼
Case 5 a1 ¼ l3 ¼ 0; l2 ¼ l2 ; a0 ¼ a0 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 l2 l21 l22 2b2 l b1 ¼ k ¼ 2 22 ; 2 2 l1 l2 l1 l2 k2 d ¼ arccosh 1 þ ; 2
a2 ¼ a1 ;
l1 ¼ l1 ;
b2 ¼ b2 :
Case 6 b1 ¼ l3 ¼ b1 ¼ 0; k ¼ 2i;
l2 ¼
l1 ¼ ib2 ; a21
a2 ¼ a1 ;
b2 ¼ b2 ;
ib2 ; a21
k2 d ¼ arccosh 1 þ : 2
a0 ¼ a0 ;
a1 ¼ a1 ;
2b2 ; k
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Case 7 b 2 ¼ b2 ; a1 ¼
l3 ¼ 0;
4ib2 ; 9b1
l1 ¼
a2 ¼ a1 ;
a0 ¼ a0 ;
b1 ¼ b1 ;
k ¼ 2i;
9ib21 ; 16b2
k2 d ¼ arccosh 1 þ : 2
9ib21 ; l2 ¼ 16b2 Case 8 a 2 ¼ a1 ; a1 ¼
l2 ¼ l1 ;
b2 ; 2l3
k ¼ k;
b1 ¼
a0 ¼ a 0 ;
b2 ¼ b2 ;
l3 ¼ l3 ;
l1 ¼ l1 ;
a0 ¼ a0 ;
a1 ¼ a1 ;
d ¼ d;
l3 b2 þ 2l21 a1 : 2l1
Case 9 a 2 ¼ a1 ; k ¼ k;
l2 ¼ l1 ; b2 ¼ l3 ¼ 0;
l1 ¼ l1 ; b1 ¼
d ¼ d;
l3 b2 þ 22l21 a1 : l1
Substituting Cases 1–9 into (9a), we can obtain the following exact solutions of Eq. (6). uðn; tÞ1 ¼ a0 þ tanhðnd þ sinhðdÞt þ dÞ; where a0, d, d are arbitrary constants. 2 2 2b þl 2 b2 sinh n arccosh 2l2 2 þ 2b t þ d l2 2 22 uðn; tÞ2 ¼ a0 þ ; 2b2 þl2 2b2 l2 cosh n arccosh l2 þ l 2 t þ d l2 2
where l2, b2, a0, d are arbitrary constants. b1 ffi pffiffiffiffiffiffiffiffi 2 2 2b2 þl2 b21 þb22 l2 2 l2 cosh n arccosh l2 t þ d þ þ 2b l2 b2 2 2 2 2b þl 2 b2 sinh n arccosh 2l2 2 þ 2b tþd l2 2 ffi ; þ pffiffiffiffiffiffiffiffi 2 2 2b þl b21 þb22 l2 2 l2 cosh n arccosh 2l2 2 þ 2b t þ d þ l2 b2
uðn; tÞ3 ¼ a0 þ
2
where l2, b2, b1, a0, d are arbitrary constants.
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2 ik k k þ 1 þ kt þ d þ uðn; tÞ4 ¼ a0 þ sech n arccosh 2 2 2 2 k tanh n arccosh þ 1 þ kt þ d ; 2 where k, a0, d are arbitrary constants. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 b2 l21 l22 uðn; tÞ5 ¼ a0 þ 2 ðl1 l22 Þl1 sinhðnn Þ þ l2 coshðnn Þ þ
b2 sinhðnn Þ ; l1 sinhðnn Þ þ l2 coshðnn Þ
2 l2 where nn ¼ n arccoshð1 þ 12 k2 Þ l2b2 l 2 t þ d and k, l1, l2, b2, a0, d are arbitrary 1 2 constants.
uðn; tÞ6 ¼ a0 þ a1 sinhðnn Þ þ a1 coshðnn Þ
ia21 sinhðnn Þ ; sinhðnn Þ coshðnn Þ
2
where nn ¼ n arccoshð1 þ k2 Þ þ 2it þ d and k, a1, d are arbitrary constants. uðn; tÞ7 ¼ a0
4ib2 sinhðnn Þ 4ib2 coshðnn Þ 16ib2 9b1 9b1 9b1 ðsinhðnn Þ coshðnn ÞÞ
16ib22 sinhðnn Þ ; 2 9b1 ðsinhðnn Þ coshðnn ÞÞ 2
where nn ¼ n arccoshð1 þ k2 Þ 2it þ d and k, b1, b2, a0, d are arbitrary constants. uðn; tÞ8 ¼ a0 þ
b2 sinhðnn Þ b2 coshðnn Þ b1 þ 2l3 2l3 l1 sinhðnn Þ l1 coshðnn Þ þ l3
b2 sinhðnn Þ ; l1 sinhðnn Þ l1 coshðnn Þ þ l3
where nn = nd + kt + d and k, l1, l3, d, b1, b2, a0, d are arbitrary constants. uðn; tÞ9 ¼ a0 þ a1 sinhðnn Þ þ a1 coshðnn Þ þ
b1 ; l1 sinhðnn Þ l1 coshðnn Þ
where nn = nd + kt + d and k, l1, d, b1, a1, a0, d are arbitrary constants. Remark. Among the above solutions, u(n, t)1, u(n, t)2, are kink-shaped solitary wave solutions. To our knowledge, all these solutions have not been given in literature.
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4. Conclusion In this paper, we presented the modified hyperbolic function method. The method is more powerful than the tanh-method proposed in [26]. The Toda lattice is chosen to illustrate the method such that nine solutions are obtained. Further work about various extension and improved hyperbolic function method needs us to find the more general ansa¨tzes.
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