Conservation laws of the (2 + 1)-dimensional KP equation and Burgers equation with variable coefficients and cross terms

In this paper, conservation laws for the (2 + 1)-dimensional KP equation and Burgers equation with variable coefficients and cross terms are studied. Due to the existence of cross terms, the “new conservation theorem” given by Ibragimov cannot be applied to the two equations directly. We propose two rules of modifications which ensure the theorem can be applied to nonlinear evolution equations with cross terms. Formulas of conservation laws for the KP equation and Burgers equation are given. Using these formulas, non-trivial and time-dependent conservation laws for these equations are derived according to their different Lie symmetries.