The extended F-expansion method and its application for a class of nonlinear evolution equations

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Abstract

By means of a simple transformation technique, we have shown that the higher-order nonlinear Schrödinger equation in nonlinear optical fibers, a new Hamiltonian amplitude equation, generalized Hirota–Satsuma coupled system and generalized ZK-BBM equation can be reduced to the elliptic-like equation. Then, the extended F-expansion method is used to obtain a series of solutions including the single and the combined nondegenerative Jacobi elliptic function solutions and their degenerative solutions to the above mentioned class of nonlinear evolution equations.

Introduction

In the recent decade, the study of nonlinear partial differential equations (NLEEs) modelling physical phenomena, has become an important toll. Seeking exact solutions for (NLEEs) has long been one of the central theme of perpetual interest in Mathematics and Physics. These solutions may well describe various phenomena in physics and other fields and thus may give more insight into the physical aspects of the problems.

As the mathematical models of complex physics phenomena, nonlinear evolution equations are involved in many fields from physics to biology, chemistry, engineer, plasma physics, optical fibers and solid state physics etc. Many methods were developed for finding the exact solutions of nonlinear evolution equations, such as Hirota’s method [1], Backlund and Darboux transformation [2], [3], [4], Painlevé expansions [5], homogeneous balance method [6], Jacobi elliptic function [7], extended tanh-function method [8], [9], F-expansion method and extended F-expansion method [10], [11], [12], [13], [14] which was proposed recently as an overall generalization of Jacobi elliptic expansion function method. Most of exact solutions were obtained by these methods, including the solitary wave solutions, shock wave solution, periodic wave solutions and so on.

Very recently, the extended F-expansion [10], [11] method has been proposed to obtain not only the single nondegenerative Jacobi elliptic function solutions, but also the combined nondegenerative Jacobi elliptic solutions and their corresponding degenerative solutions.

In this paper, we will use the extended F-expansion method to construct exact solutions to some class of nonlinear evolution equations which can be reduced to a simple elliptic-like equation. Applying this method, we have successfully found the solutions of higher-order nonlinear Schrödinger equation, a new Hamiltonian amplitude equation, generalized Hirota–Satsuma coupled Kdv system and ZK-BBM equation.

Section snippets

Extended F-expansion method

Considering a give nonlinear partial differential equation with independent variables x = (x1, x2, x3,  , xl, t), and dependent variables u, the unknowns u = (x1, x2, x3,  , xl, t) are solutions of the NODE obtained by the travelling wave reduction (x1, x2, x3,  , xl, t)  u(ξ = λ1x1 + λ2x2 + λ3x3 +  + λlxl  ωt). Thus, we get a NODE for u(ξ) as follows:F(u,u,u,)=0.

To determine u(ξ) explicitly, we take the following four steps:

  • Step (1):

    Supposing that u(ξ) can be expanded as follows:u(ξ)=j=0ni=0jcjiFi(ξ)Gj-i(ξ),cnn0oru(ξ)=a0+i=1n

Solutions of the elliptic-like equation

Let us consider the following elliptic-like equation:Aϕ(ξ)+Bϕ(ξ)+Dϕ3(ξ)=0,where A, B and D are arbitrary constants.

Considering the homogeneous balance between ϕ″(ξ) and ϕ3(ξ) we get n = 1, so the solution of (7) is the formϕ(ξ)=a0+a1F(ξ)+b1G(ξ),where a0, a1 and b1 are constants to be determined, F(ξ) and G(ξ) satisfy NODE (4), (5), (6). Substituting (8) into (7) along with (4), (5), (6) and collecting coefficients of the Fp(ξ)Gq (p = 0, 1, 2, 3; q = 0, 1) we have the following algebraic equations:F0:a0[B

Applications

In this section, by using the results obtained in the preceding section, we will construct the corresponding solutions of some class of nonlinear evolution equations of special interest in mathematical physics such as the higher-order nonlinear Schrödinger equation in nonlinear optical fibers, a new Hamiltonian amplitude equation, generalized Hirota–Satsuma coupled system and generalized ZK-BBM equation.

Example 4.1

The higher-order nonlinear Schrödinger equation

The higher-order nonlinear Schrödinger

Conclusions

By introducing appropriate transformations and using extended F-expansion method, we have been able to obtain in a unified way with the aid of symbolic computation system-Maple, a series of solutions including single and the combined nondegenerative Jacobi elliptic function solutions and their degenerative solutions to a some class of nonlinear evolution equations of special interest in mathematical physics. This class of NLPDES is characterized by the fact that it can be reduced with the aid

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