Global well-posedness and standing wave solutions for a class of nematic liquid crystal system

Abstract

This paper is devoted to studying the Cauchy problem and standing wave for a class of nematic liquid crystal system. This system appears in the recent studies of the propagation of a laser beam in a nematic liquid crystal. The above system couples a Schr\"{o}dinger evolution equation to a nonlinear elliptic equation which describes the response of the director angle to the laser beam electric field. The global well-posedness will be established through using Banach fixed point theorem and continuity argument. Secondly, the existence of standing wave solution is established through using the constrained minimization approach.