Lie Symmetries, One-dimensional Optimal System And Optimal Reduction Of (2+1)-coupled Nonlinear Schr(o|")edinger Equations

As well known, a lot of significant natural science and Engineering and technical problems are come down to in research of nonlinear partial differential equations. The exact solutions of the nonlinear partial differential equations have important value in the theory and applications described by the equations. The exact solutions give good ways to explain variety of natural phenomena, such as vibration and soliton wave propagation etc. Since the well development of inverse scattering method, especially as the development of many symbolic computation software, such as Mathematica, Matlab, Maple etc have promoted the study of nonlinear partial differential equations in recent. Therefor the study of solving exact solutions of nonlinear partial differential equations gradually becomes a very active topic in the field of differential equations and have being attracted widespread attentions of researchers. However, even there many methods have been put forward for solving nonlinear partial differential equations, they have own disadvantages. So looking for some effective and feasible method is still a very important and valuable work to study.In this paper, we studied symmetry reduction method of nonlinear differential e-quations which proposed by S on a class of (1+2)-dimensional nonlinear Schroedinger equations. We obtain the following results:the infinite dimensional Lie algebra of the classical symmetry group of the equations is found; the one-dimensional optimal system of an8-dimensional subalgebra of the infinite Lie algebra is constructed; the reduced e- quations of the equations with respect to the optimal system are derived. Furthermore, the one-dimensional optimal systems of the Lie algebra admitted by the reduced equa-tions are also constructed. Consequently, the classification of the twice optimal symmetry reductions of the original equations with respect to the optimal systems are presented. The reductions show that the (1+2)-dimensional nonlinear Schroedinger equations can be reduced to a group of ordinary differential equations which is useful for solving the related problems of the equations.