Exact solution of a KdV equation with variable coefficients

The modified KdV equation with variable coefficients: Exact uni/bi-variable travelling wave-like solutions

In this paper, the modified Korteweg-de Vries (mKdV) equation with variable coefficients (vc-mKdV equation) is investigated via two kinds of approaches and symbolic computation. On the one hand, we firstly reduce the vc-mKdV equation to a second-order nonlinear nonhomogeneous ODE using travelling wave-like similarity transformation. And then we obtain its many types of exact fractional solution...

Exact Solutions for a Generalized KdV-MKdV Equation with Variable Coefficients

By using solutions of an ordinary differential equation, an auxiliary equationmethod is described to seek exact solutions of variablecoefficient KdV-MKdV equation. As a result, more new exact nontravelling solutions, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions, and combined Jacobi elliptic function solutions, for...

Exact Solutions for a Third-Order KdV Equation with Variable Coefficients and Forcing Term

New Exact Solutions of a Variable-Coefficient KdV Equation

In this paper, we apply the Miura transformation to construct the connection between a variablecoefficient KdV (vcKdV) equation and a variable-coefficient modified KdV (vcmKdV) equation under certain constraint. Solving the vcmKdV equation by use of the auxiliary equation method and using the Miura transformation, we find a rich variety of new exact solutions for the vcKdV equation, which inclu...

Order reduction and μ-conservation law for the non-isospectral KdV type equation a with variable-coefficients

The goal of this paper is to calculate of order reduction of the KdV typeequation and the non-isospectral KdV type equation using the μ-symmetrymethod. Moreover we obtain μ-conservation law of the non-isospectral KdVtype equation using the variational problem method.