Lie group analysis of Schr¨odinger equations describing optical waves in birefringent waveguides

Abstract

Lie group analysis will be carried out for a system of two nonlinear Schr¨odinger equations characterizing the propagation of optical pulses and involving fourwave mixing terms in a birefringent media. The ultimate goal is to find the most general symmetry transformation that preserves the solution space and generates in particular a series of solutions of the system starting from a seed solution. Such a general symmetry transformation is the symmetry group, and it will be computed from the symmetry algebra L of the system rewritten as a system of four nonlinear uncoupled equations in real form. An optimal system of one dimensional subalgebras of L will be found and some related symmetry reductions will be given. Soliton or other solutions will be obtained by means of either the Hirota direct method, or some substitution methods such as the Tanh-expansion method or its variants else by other common methods including the direct Lie group methods.