Several Groups Of (1 +1)-dimensional (2 +1)-dimensional Soliton Equations, Two Types Of Darboux Transformations And Exact Solutions,

The solitonian theory is an important part of the nonlinear science. There are many nonlinear partial differential equations that have solition properties in the pure and applied sci-ence.Therefore,solving soliton equations (especial (2+1) dimensional equation) is important in the field. There are many methods to obtain explicit solutions of nonlinear partial differential equations in the solition theory. Among the various approaches, the Darboux transformation is a very powerful tool,it can find explict solutions of solition equations from a trivial seed. The Darboux transformation of two (1+1)-dimensional soliton equations and (2+1)-dimensional KP equation is obtained in this paper.There are three sections in the paper. The fist part is an introduction.In the second section,we directly construct the N-fold Darboux transformtion of soliton equation. In fact,it is a gauge transformationφ|- = Tφ,withof Lax pairsIfφ|- satisfies the Lax pairs , then,the new solutions (u|-, v|-)of soliton equation can be obtained as follow through DT:where (u, v) are solutions of two (1+1)-dimensional soliton equations and (2+1)-dimensional KP equation .The exact solutions of soliton equation are obtained by means of the Darboux transformation;and several interesting figures of the solutions are plotted. In Section Three,we directly construct the N-fold Darboux transformtion of soliton equation. In fact,it is a gauge transformationφ|- = Tφ,withof Lax pairsIfφ|- satisfies the Lax pairs , then,the new solutions (u|-,v|-)of soliton equation can be obtained as follow by DT:where (u,v) are solutions of two (1+1)-dimensional soliton equations and (2+1)-dimensional KPequation .The exact solutions of soliton equation are obtained by means of the Darboux transformation;andseveral interesting figures of the solutions are plotted.