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

Abstract

In this paper, conservation laws for the $(\text{2}+\text{1})$-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.