A New Variable-Coefficient Riccati Subequation Method for Solving Nonlinear Lattice Equations

Fanwei Meng

doi:10.1155/2013/810363

Abstract

We propose a new variable-coefficient Riccati subequation method to establish new exact solutions for nonlinear differential-difference equations. For illustrating the validity of this method, we apply it to the discrete (2 + 1)-dimensional Toda lattice equation. As a result, some new and generalized traveling wave solutions including hyperbolic function solutions, trigonometric function solutions, and rational function solutions are obtained.

Highlights

Nonlinear differential-difference equations (NDDEs) play an important role in many branches of applied physical sciences such as condensed matter physics, biophysics, atomic chains, molecular crystals, and quantum physics
Since the work of Fermi in the 1960s [1], research for NDDEs has been paid a lot of attention recently
We notice most of the existing methods are dealing with ansatzes with constant coefficients, while very few methods are concerned with variable coefficients

Summary

Introduction

Nonlinear differential-difference equations (NDDEs) play an important role in many branches of applied physical sciences such as condensed matter physics, biophysics, atomic chains, molecular crystals, and quantum physics. The investigation of exact solutions of NDDEs has attracted a wide interest, and many effective methods have been presented and applied for solving NDDEs successfully in the literature. These methods include the known (G󸀠/G)-expansion method [11,12,13,14], the exp-function method [15], the exponential function rational expansion method [16, 17], the Jacobi elliptic function method [18, 19], Hirota’s bilinear method [20], the extended simplest equation method [21], and the tanh-function method [22].

Description of the Variable-Coefficient Riccati Subequation Method

Conclusions