A discrete generalization of the extended simplest equation method. (English) Zbl 1222.65114

Summary: We modify the so-called extended simplest equation method to obtain discrete traveling wave solutions for nonlinear differential-difference equations. The Wadati lattice equation is chosen to illustrate the method in detail. Further discrete soliton/periodic solutions with more arbitrary parameters, as well as discrete rational solutions, are revealed. We note that using our approach one can also find in principal highly accurate exact discrete solutions for other lattice equations arising in the applied sciences.


65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
35C07 Traveling wave solutions
34A33 Ordinary lattice differential equations
37K60 Lattice dynamics; integrable lattice equations
Full Text: DOI Link


[1] Fermi, E.; Pasta, J.; Ulam, S., Collected papers of enrico Fermi II, (1965), University of Chicago Press Chicago, IL, p. 978
[2] Scott, A.C.; Macheil, L., Binding energy versus nonlinearity for a small stationary soliton, Phys lett A, 98, 87-88, (1983)
[3] Su, W.P.; Schrieffer, J.R.; Heege, A.J., Solitons in polyacetylene, Phys rev lett, 42, 1698-1701, (1979)
[4] Davydov, A.S., The theory of contraction of proteins under their excitation, J theor biol, 38, 559-569, (1973)
[5] Marquii, P.; Bilbault, J.M.; Rernoissnet, M., Observation of nonlinear localized modes in an electrical lattice, Phys rev E, 51, 6127-6133, (1995)
[6] Hu, X.B.; Ma, W.X., Application of hirota’s bilinear formalism to the Toeplitz lattice—some special soliton-like solutions, Phys lett A, 293, 161-165, (2002) · Zbl 0985.35072
[7] Liu, S.K.; Fu, Z.T.; Wang, Z.G.; Liu, S.D., Periodic solutions for a class of nonlinear differential-difference equations, Commun theor phys, 49, 1155-1158, (2008) · Zbl 1392.34085
[8] Baldwin, D.; Goktas, U.; Hereman, W., Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations, Comput phys commun, 162, 203-217, (2004) · Zbl 1196.68324
[9] Dai, C.Q.; Meng, J.P.; Zhang, J.F., Symbolic computation of extended Jacobian elliptic function algorithm for nonlinear differential-different equations, Commun theor phys, 43, 471-478, (2005)
[10] Zhu, S.D., Exp-function method for the hybrid-lattice system, Int J nonlinear sci, 8, 461-464, (2007)
[11] Zhu, S.D., Exp-function method for the discrete mkdv lattice, Int J nonlinear sci, 8, 465-469, (2007)
[12] Xie, F.; Jia, M.; Zhao, H., Some solutions of discrete sine-Gordon equation, Chaos solitons fract, 33, 1791-1795, (2007) · Zbl 1129.35456
[13] Yang, P.; Chen, Y.; Li, Z.B., ADM-Padé technique for the nonlinear lattice equations, Appl math comput, 210, 362-375, (2009) · Zbl 1162.65399
[14] Zhu, S.D.; Chu, Y.M.; Qiu, S.L., The homotopy perturbation method for discontinued problems arising in nanotechnology, Comput math appl, (2009) · Zbl 1189.65186
[15] Zhen, W., Discrete tanh method for nonlinear difference-differential equations, Comput phys comm, 180, 1104-1108, (2009) · Zbl 1198.65157
[16] Kudryashov, N.A., Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos solitons fract, 24, 1217-1231, (2005) · Zbl 1069.35018
[17] Kudryashov, N.A., Exact solitary waves of the Fisher equation, Phys lett A, 342, 99-106, (2005) · Zbl 1222.35054
[18] Kudryashov, N.A.; Loguinova, N.B., Extended simplest equation method for nonlinear differential equations, Appl math comput, 205, 396-402, (2008) · Zbl 1168.34003
[19] Wadati, M., Transformation theories for nonlinear discrete systems, Prog theor phys suppl, 59, 36-63, (1976)
[20] Adler, V.E.; Svinolupov, S.I.; Yamilov, R.I., Multi-component Volterra and Toda type integrable equations, Phys lett A, 254, 24-36, (1999) · Zbl 0983.37082
[21] Ablowitz, M.J.; Ladik, J.F., On the solution of a class of nonlinear partial difference equations, Stud appl math, 57, 1-12, (1977) · Zbl 0384.35018
[22] Dai, C.; Cen, X.; Wu, S., The application of he’s exp-function method to a nonlinear differential-difference equation, Chaos solitons fract, (2008)
[23] Xie, F.; Wang, J., A new method for solving nonlinear differential-difference equation, Chaos solitons fract, 27, 1067-1071, (2006) · Zbl 1094.34058
[24] Wu, G.; Xia, T., A new method for constructing soliton solutions to differential-difference equation with symbolic computation, Chaos solitons fract, 39, 2245-2248, (2009) · Zbl 1197.35250
[25] Kudryashov, N.A.; Loguinova, N.B., Be careful with the exp-function method, Commun nonlinear sci numer simul, 14, 1881-1890, (2009) · Zbl 1221.35344
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.