The determining equations for the nonclassical method of the nonlinear differential equation(s) with arbitrary order can be obtained through the compatibility

Abstract

In this paper, firstly we show that the determining equations of the ( 1 + 1 ) dimension nonlinear differential equation with arbitrary order for the nonclassical method can be derived by the compatibility between the original equation and the invariant surface condition. Then we generalize this result to the system of the ( m + 1 ) dimension differential equations. The nonlinear Klein–Gordon equation, the ( 2 + 1 ) -dimensional Boussinesq equation and the generalized Nizhnik–Novikov–Veselov equation serve as examples illustrating this method.