Conservation Laws, Symmetry Reductions, and New Exact Solutions of the (2 + 1)-Dimensional Kadomtsev-Petviashvili Equation with Time-Dependent Coefficients

Li-Hua Zhang

doi:10.1155/2014/853578

Abstract

The (2 + 1)-dimensional Kadomtsev-Petviashvili equation with time-dependent coefficients is investigated. By means of the Lie group method, we first obtain several geometric symmetries for the equation in terms of coefficient functions and arbitrary functions oft. Based on the obtained symmetries, many nontrivial and time-dependent conservation laws for the equation are obtained with the help of Ibragimov’s new conservation theorem. Applying the characteristic equations of the obtained symmetries, the (2 + 1)-dimensional KP equation is reduced to (1 + 1)-dimensional nonlinear partial differential equations, including a special case of (2 + 1)-dimensional Boussinesq equation and different types of the KdV equation. At the same time, many new exact solutions are derived such as soliton and soliton-like solutions and algebraically explicit analytical solutions.

Highlights

The Lie group method is a powerful tool to perform Lie symmetry analysis, study conservation laws, and look for exact solutions of nonlinear partial differential equations (NLPDEs) [1,2,3,4]
We first construct the conservation laws for the system consisting of the initial equation (1) and its adjoint (28)
Using the obtained symmetries (3), similarity variables and symmetry reductions can be found by solving the corresponding characteristic equation: dx ξ

Summary

Introduction

The Lie group method is a powerful tool to perform Lie symmetry analysis, study conservation laws, and look for exact solutions of nonlinear partial differential equations (NLPDEs) [1,2,3,4]. Where f(t) and g(t) are arbitrary functions, C3 is an integral constant, and n(t) and τ(t) satisfy the following ordinary differential equation: nttt This shows that, under the condition (19), the equation uxt + 6ux2 + 6uuxx + uxxxx We first construct the conservation laws for the system consisting of the initial equation (1) and its adjoint (28). For the symmetry in Case 2, the corresponding components of the conservation laws are

C2yuyuxV p

Conclusions