Conservation Laws of Two(2+1)-Dimensional Nonlinear Evolution Equations with Higher-Order Mixed Derivatives

Li-Hua Zhang

doi:10.1155/2013/594975

Abstract

In this paper, conservation laws for the(2+1)-dimensional ANNV equation and KP-BBM equation with higher-order mixed derivatives are studied. Due to the existence of higher-order mixed derivatives, Ibragimov’s “new conservation theorem” cannot be applied to the two equations directly. We propose two modification rules which ensure that the theorem can be applied to nonlinear evolution equations with any mixed derivatives. Formulas of conservation laws for the ANNV equation and KP-BBM equation are given. Using these formulas, many nontrivial and time-dependent conservation laws for these equations are derived.

Highlights

The construction of explicit forms of conservation laws plays an important role in the study of nonlinear science, as they are used for the development of appropriate numerical methods and for mathematical analysis, in particular, existence, uniqueness, and stability analysis [1,2,3]
As shown in [16], when applying Noether’s theorem and partial Noether’s theorem to obtain conservation laws of nonlinear evolution equations with higher-order mixed derivatives, the obtained conservation laws must be adjusted to satisfy the definition of conservation laws
We propose two modification rules which ensure that Ibragimov’s theorem can be applied to nonlinear evolution equations with higher-order and lower-order mixed derivatives

Summary

Introduction

The construction of explicit forms of conservation laws plays an important role in the study of nonlinear science, as they are used for the development of appropriate numerical methods and for mathematical analysis, in particular, existence, uniqueness, and stability analysis [1,2,3]. For any linear or nonlinear differential equations, Ibragimov’s new conservation theorem offers a procedure for constructing explicit conservation laws associated with the known Lie, Lie-Backlund, or nonlocal symmetries. It does not require the existence of classical Lagrangians. In [9], we have proposed two modification rules to apply Ibragimov’s theorem to study conservation laws of two evolution equations with mixed derivatives, but the mixed derivatives are all second order and not the highest derivative term.

Preliminaries

Two Modification Rules and Conservation Laws for the ANNV Equation

Concluding Remarks