(2 +1) - Dimensional Dispersive Long Wave System Symmetry Reduction

When one is confronted with a complicated system of partial differential equations arising from some physically important problems, the discovery of any explicit solution whatsoever is of great interest. Explicit solutions can be used as models for physical experiments, as benchmarks for testing numerical methods, etc., and often reflect the asymptotic or dominant behaviour of more general types of solutions. The methods used to find group-invariant solutions, generalizing the well-known techniques for finding similarity solutions, provides a systematic computational method for determining large classes of special solutions. These group-invariant solutions are characterized by their invariance under some symmetry group of the system of partial differential equations; the more symmetrical the solution, the easier it is to construct. Roughly speaking, the solutions which are invariant under a given r-parameter symmetry group of the system can all be found by solving a system of differential equations involving r fewer independent variables than original system. In this way, one reduces an intractable set of partial differential equations to a simpler set of partial differential equations, or even to a set of ordinary differential equations, which one might stand a chance of solving explicitly. In this paper, by applying this method, we study the (2+1)-dimensional dispersive long-wave system. Recently, many straightforward and effective methods have been devoted to study of this system ([6] , [7], [21], [24-25], [27-30] ).But most papers in the literature propose some kinds of ansatz, that is, first assuming that the solutions are in some special forms and then trying to determine all the unknown functions. We get the group-invariant solutions of this system via symmetry method, avoiding the special form of the solutions. First, we determine the symmetry group of the system. It turns out that the symmetry group contains three arbitrary smooth functions. Then by using some subgroups, we reduce the system to the heat equation and the second Painlevéequation. In some cases, we solve the reduced equations explicitly.