Darboux Transformations And Exact Solutions Of (2+1) Dimensional Soliton Equations

In this dissertation,under the guidance of mathematical mechanization and the AC=BD theory put forward by Prof.Zhang Hongqing.and by means of symbolic computation software Maple,some topics in symbolic integration and differential equation are studied. Among the various approaches,the Darboux transformation is a very powerful tool,which can be used to find explicit solutions of soliton equations from a trivial seed.Chapter 1 is to introduce the related development of mathematical physics mechanization, emphasizing on the relation between differential equations and computer algebra.We give an introduction of mathematical physics mechanization at home and abroad in summary.Chapter 2 concerns the construction of transformation of differential equations under the uniform frame work of AC=BD model theory introduced by Prof.H.Q.Zhang.The basic theory of C-D pair is presented.Chapter 3 is to introduce the history and theory of Darboux transformation.In chapter 4 and 5,we take Levi KPII and mKP soliton equations for example,based on the gauge transformation of the spectral problem,we obtain the Darboux transformations of some(2+1) dimensional soliton equations and use the Darboux transformation for generating the exact solutions of the(2+1) dimensional soliton equations.