An algorithmic method for showing existence of nontrivial non-classical symmetries of partial differential equations without solving determining equations

In this paper, based on differential characteristic set theory and the associated algorithm (also called Wuʼs method), an algorithmic method is presented to decide on the existence of a nontrivial non-classical symmetry of a given partial differential equation without solving the corresponding nonlinear determining system. The theory and algorithm give a partial answer for the open problem posed by P.A. Clarkson and E.L. Mansfield in [21] on non-classical symmetries of partial differential equations. As applications of our algorithm, non-classical symmetries and corresponding invariant solutions are found for several evolution equations.