A multiple exp-function method for nonlinear differential equations and its application

Abstract

A multiple exp-function method to exact multiple wave solutions of nonlinear partial differential equations is proposed. The method is oriented towards ease of use and capability of computer algebra systems, and provides a direct and systematical solution procedure which generalizes Hirota's perturbation scheme. With help of Maple, an application of the approach to the 3 + 1 dimensional potential-Yu-Toda-Sasa-Fukuyama equation yields exact explicit 1-wave and 2-wave and 3-wave solutions, which include 1-soliton, 2-soliton and 3-soliton type solutions. Two cases with specific values of the involved parameters are plotted for each of 2-wave and 3-wave solutions.

Exact and explicit solutions to the discrete nonlinear Schrödinger equation with a saturable nonlinearity

“…It seems that the multiple Exp-function method [25] is more general than the (G /G)-expansion method. The adopted (G /G)-expansion method actually uses the same idea as the one using an integrable ODE v ξ = av + bv 2 with constants a and b, which was presented for the first time in [31].…”

“…Nowadays, some modern analytic methods are available for analyzing nonlinear evolution-type equations. For instance, tanh function method [20], homotopy perturbation method [21], variational iteration method [22], first integral method [23], Expfunction method [24], multiple Exp-function method [25], linear superposition principle [26], Wronskian technique [27], the Fexpansion method [28], the Jacobi elliptic-function method [29], the similarity transformation method [30], the ansätze method [31]. On the other hand, since it was first enunciated in 2008 by 0375-9601/$ -see front matter © 2011 Elsevier B.V. All rights reserved.…”

Exact Traveling Wave Solutions and Simulation For A Class Nonlinear Partial Differential Equations

“…There are many methods to be used in travel wave solutions of nonlinear evolution equations, such as the inverse scattering method, the B a  cklund transformation method, algebraic-geometric method, the Darboux transformation method, multiple exp-function method [4] , the Hirota bilinear method [2,3,8] , and dynamical systems method . By applying the dynamical systems method [5,6,7], some new exact solutions of (1.1) are obtained [7] .…”

Explicit Analytic Solution for the Nonlinear Evolution Equations using the Simplest Equation Method

“…So we search for a mathematical algorithm to discover the exact solutions of nonlinear partial differential equations. Many papers has been focused on the application of the known methods to construct the solutions of nonlinear evolution equation (NLEE), among them the inverse scattering transform [ [1]- [3]] and Hirota method [4], tanh method [1,2,3,4,5,6], multiple exp-function method [7], Backlund transformation method [8], simplest equation method [9,10], and so on.…”