Qualitative Analysis Of Solitary Waves In Nematic Liquid Crystals

In this paper,we consider the qualitative analysis of solitary waves in nematic liquid crystals.Using genus theorem and Nehari manifold techniques,we prove the existence and radial symmetry of soliton solutions in(2+1)dimensional nonlocal nematic liquid crystals.Based on the concentration compactness principle,we prove the existence and orbital stability of the ground state solitary waves in nonlocal nematic liquid crystals.This paper is divided into four parts.In Chapter 1,we introduce the background,the development,and the main results in this paper.In Chapter 2,we investigate some properties of soliton solutions in(2+1)dimensional nonlocal nematic liquid crystals.Based on the relative category theory and Nehari manifold techniques,we prove the existence of high energy soliton solutions with arbitrarity many nodal domains,and the ground state soliton solutions which has fixed sign.Hence,without loss of generality,we obtain the existence of positive ground state soliton solutions.Furthermore,we also prove the uniqueness and radial symmetry of positive soliton solutions.So,we present the uniqueness and radial symmetry of ground state soliton solutions.In Chapter 3,based on fixed point argument in space-time Lebesgue space,Gagliardo-Nirenberg inequality and concentration compactness principle,we obtain the existence and orbital stability of the ground state solitary waves with sufficiently large L~2-norm in nematic liquid crystals.In Chapter 4,using genus theorem and Nehari manifold techniques in critical points theory,we prove the existence of high energy solutions for a class of Schr(5)o(5)dinger-Poissonsystems with periodic potential in dimension two,and obtain that the solution has infinitely nodal domains.Furthermore,the existence of ground state solution is proved.The Schr(5)o(5)dinger-Poissonsystems is the generalization of nematic liquid crystals model systems.