Abstract
It is well known that different types of exact solutions of an auxiliary equation produce new types of exact travelling wave solutions to nonlinear equations. In this paper, by means of symbolic computation, the new solutions of original auxiliary equation of first-order nonlinear ordinary differential equation with a sixth-degree nonlinear term are presented to obtain novel exact solutions of the Kawahara equation. By the aid of the solutions of the original auxiliary equation, some other physically important nonlinear equations can be solved to construct novel exact solutions.
Highlights
Over the past decades a number of approximate methods for finding travelling wave solutions to nonlinear evolution equations have been proposed or developed and modified
One of the current methods is so called auxiliary equation method [20,21,22,23,24]. The technique of this method consist of the solutions of the nonlinear evolution equations such that the target solutions of the nonlinear evolution equations can be expressed as a polynomial in a linearly independent elementary function which satisfies a particular ordinary differential equation which is named as auxiliary equation in general
Substituting the above coefficients into ansatz (26) with the solution (33) of auxiliary equation, we obtain another new solution of Kawahara equation: u (x, t)
Summary
Over the past decades a number of approximate methods for finding travelling wave solutions to nonlinear evolution equations have been proposed or developed and modified. References [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24] and references within cite some of this accomplishment Among these methods, one of the current methods is so called auxiliary equation method [20,21,22,23,24]. We will examine the consequences of the choice of the auxiliary equation for determining the solutions of the nonlinear evolution equation in consideration and to seek more types of new exact solutions of nonlinear differential equations which satisfy a first-order nonlinear ordinary differential equation with a sixth-degree nonlinear term
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