Algorithmic framework for group analysis of differential equations and its application to generalized Zakharov--Kuznetsov equations

摘要

In this paper, we explain in more details the modern treatment of the problemof group classification of (systems of) partial differential equations (PDEs)from the algorithmic point of view. More precisely, we revise the classicalLie--Ovsiannikov algorithm of construction of symmetries of differentialequations, describe the group classification algorithm and discuss the processof reduction of (systems of) PDEs to (systems of) equations with smaller numberof independent variables in order to construct invariant solutions. The groupclassification algorithm and reduction process are illustrated by the exampleof the generalized Zakharov--Kuznetsov (GZK) equations of form$u_t+(F(u))_{xxx}+(G(u))_{xyy}+(H(u))_x=0$. As a result, a complete groupclassification of the GZK equations is performed and a number of newinteresting nonlinear invariant models which have non-trivial invariancealgebras are obtained. Lie symmetry reductions and exact solutions for twoimportant invariant models, i.e., the classical and modifiedZakharov--Kuznetsov equations, are constructed. The algorithmic framework forgroup analysis of differential equations presented in this paper can also beapplied to other nonlinear PDEs.