Nonclassical symmetry reductions of nonlinear partial differential equations

Abstract

Some nonclassical symmetry reductions and exact solutions for several physically and mathematically significant, nonlinear partial differential equations are presented. These are obtained using the direct method, which involves no group theoretical techniques, originally developed by Clarkson and Kruskal [1] to study symmetry reductions of the Boussinesq equation; in fact, they are not obtainable using the classical Lie method for finding group-invariant solutions of partial differential equations. Examples of equations discussed will include the Boussinesq and Kadomtsev-Petviashvili equations, which are completely integrable soliton equations, and multidimensional cubic, quintic and derivative nonlinear Schrödinger equations, the Fitzhugh-Nagumo equation, the Navier-Stokes equation and the Zabolotskaya-Khokhlov (two-dimensional Burgers) equation, which are thought not to be completely integrable.