| Soliton theory is an important research area in nonlinear science. In this dissertation, three important nonlinear equations are mainly discussed by using the Darboux transformation and the multi-linear variable separation approach.This dissertation is devided into five chapters.In chapter I, the discovery of solitons and its developments are reviewed by briefly. We also introduced the original Darboux transformation, the Darboux matrix approach and the multi-linear variable separation approach.In chapter II, we discussed Darboux transformation of Kaup-Kupersch- midt equation and generated the new soliton solutions of Kaup-Kuperschmidt equation by using the Darboux transformation from the trivial solution u1 =0.In chapter III , the new Lax pair of CDGKS equation is presented.In chapter IV, we discussed the multi-linear variable separation approach of the (2 +1)-dimensional dispersive long wave equations and a new general variable separated solutions for (2 +1)-dimensional disper- sion long wave equations in the from is obtained by use of the truncated Painlevéexpansion approach.In this chapter, the multi-linear variable separation approach of the (2+1)-dimensional dispersion long wave equations with variable coefficients is discussed. A new general variable separated solutions in the from is obtained by using the truncated Painlevéexpansion approach.In chapter V, we summarized the contents of this paper. We also discuss- ed main and important results as well as the research topics of furture in our work. |
