Exponential function rational expansion method for nonlinear differential–difference equations

Abstract

An exponential function rational expansion method for solving exact traveling wave solutions to nonlinear differential–difference equations is proposed. By this method, new exact traveling wave solutions to some nonlinear differential–difference equations such as Langmiuir lattice, discrete mKdV lattice equation, Hybrid lattice equation, etc., are obtained.

Introduction

Differential–difference equations (DDEs) play a crucial role in modelling of many physical phenomena [1], [2] such as particle vibrations in lattices, currents in electrical networks, pulses in biological chains, etc. Unlike difference equations which are fully discretized, differential–difference equations are semi-discretized with some (or all) of their special variables discretized while time is usually kept continuous. In [3], [4], [5], [6], [7], [8], some aspects of DDEs, such as integrability criteria, the computation of densities and symmetries, conservation laws and transformation theory, etc., have been studied extensively. In the theory of solitons, as we know, the inverse scattering method (ISM) is one of the most useful tools to solve various integral models. Flaschka [9], Manakov [10], Ablowitz and Ladik [11], [12], and Tsuchida, Ujino and Wadadi [13], [14] showed that the discrete version of the inverse scattering method was also applicable to discrete integrable models such as the Toda lattice, the Volterra model, the discrete mKdV equation, the discrete NLS equation, the semi-discrete coupled modified KdV equation, a system of semi-discrete coupled nonlinear Schrödinger equations, etc.

On the other hand, in order to find directly exact solutions to nonlinear DDEs, some methods [15], [16], [17], [18], [19], [20], [21], [22], [23] for solving nonlinear differential equations are applied to DDEs. For example, Qian, Lou and Hu [15] have extended successfully multilinear variable separation approach to a special Toda lattice equation. Baldwin, Göktas and Hereman [16] and Zhu [17] have applied tanh-method and its generalization to DDEs, and hence obtained some exact solutions to a lot of differential–difference equations. Dai and Zhang [18] have given a Jacobian elliptic function expansion method to solve the doubly periodic traveling wave solutions and kink-type tanh solitary solutions to some DDEs.

In this paper, for directly finding the traveling wave solutions to nonlinear DDEs, we propose an exponential function rational expansion method. This method unifies the tanh-method and its generalization. For illustration, we apply the proposed method to some nonlinear differential–difference equations such as Langmiuir lattice, discrete mKdV lattice equation, Hybrid lattice equation, etc., and obtain many exact solutions to them. Among those, to our knowledge, some solutions are new.

This paper is organized as follows: In Section 2, we present an exponential function rational expansion method for solving the solutions to differential–difference equations. In Section 3, we give some applications of the proposed method. The last section contains the conclusions and discussions. Moreover, an open problem is also discussed in this section.