Initial-value problems for evolutionary partial differential equations and higher-order conditional symmetries

Abstract

We suggest a new approach to the problem of dimensional reduction of initial/boundary value problems for evolution equations in one spatial variable. The approach is based on higher-order (generalized) conditional symmetries of the equations involved. It is shown that reducibility of an initial value problem for an evolution equation to a Cauchy problem for a system of ordinary differential equations can be fully characterized in terms of conditional symmetries which leave invariant the equation in question. We also give some examples of the solution of initial value problems for second- and third-order nonlinear differential equations by reduction by their conditional symmetries. We give a systematic classification of general second-order partial differential equations admitting second-order conditional symmetries, based on Lie’s classification of invariant second-order ordinary differential equations. This yields five classes of principally new initial value problems for nonlinear evolution equations which admit no Lie symmetries and are reducible via second-order conditional symmetries.