Exact Traveling Wave Solutions for Generalized Camassa-Holm Equation by Polynomial Expansion Methods

Abstract

We formulate efficient polynomial expansion methods and obtain the exact traveling wave solutions for the generalized Camassa-Holm Equation. By the methods, we obtain three types traveling wave solutions for the generalized Camassa-Holm Equation: hyperbolic function traveling wave solutions, trigonometric function traveling wave solutions, and rational function traveling wave solutions. At the same time, we have shown graphical behavior of the traveling wave solutions.

New exact solutions for Kudryashov–Sinelshchikov equation

“…For VIII, because μ = - 1 16 ν 2 , λ = -1 4 ν, thus q = 4νλ 2 = - 5 where ξ = x + 5ν 2 16 t and ν, k 2 , k 21 are arbitrary constants. For IX, because μ = 0, λ = - 1 4 ν, thus q = 4νλ 2 = -1 16 ν 2 < 0, then where ξ = x -3ν 16 t and ν, k 2 , k 21 are arbitrary constants.…”

“…Remark: In [5,[33][34][35][36][37], the solutions "G" of the auxiliary equation are directly obtained according to the method of solving the second ordinary differential equation, and by the expression of G, the authors obtained the expression of G G . In [33][34][35][36][37], there are only two kinds of solutions G corresponding to the discriminant of the second ordinary differential equation which is larger than zero and less than zero, respectively; and in [5], the authors obtain three kinds of solutions corresponding to the discriminant of the second ordinary differential equation which is larger than zero, equal to zero and less than zero, respectively. However, it is difficult to solve the second order ordinary differential equation, sometimes it cannot get the simple expression of its solution or cannot obtain its solutions.…”

“…So, Eq. (1.1) is a generalization of the KdV equation and the KdVB equation and it is similar but not identical to the Camassa-Holm (CH) equation (see [5] and the references therein). It is well known that pressure waves in a gas-liquid mixture is characterized by the KdVB equation and KdV equation [3,6].…”

“…Seadawy et al [32] proposed the sech-tanh method to solve the Olver equation and the fifth-order KdV equation and obtained traveling wave solutions; in [33][34][35][36][37][38] was introduced a method called the G G -expansion method and one obtained a traveling solution for the four well established nonlinear evolution equations. In [5], the authors obtained traveling wave solutions for the generalized Camassa-Holm equation by polynomial expansion methods. Those methods are very efficient, reliable, simple in solving many PDEs.…”

“…In this paper, on the basis of the G G -expansion method [5,[33][34][35][36][37][38], we use the solutions of the Riccati equation [39] to extend the auxiliary equation G + λG + μG = 0 and obtain more exact solutions of the auxiliary equation [40], thus we derive more new exact solutions of Eq. (1.1).…”