Wronskian, Grammian And Pfaffian Technique Of Nonlinear Evolution Equations

The Wronskian, Grammian and Pfaffian technique is a powerful tool to construct multisoliton solutions for many nonlinear evolution equations possessing Hirota bilinear forms. In the process of utilizing Wronskian, Grammian and Pfaffian technique, the main difficulty lies in the construction of a system of linear differential conditions, which are not unique. In this paper, we give a unique method to a system of linear differential conditions. Especially, with the help of the properties of the computing of Young diagram, we have first applied Young diagram proved the proposition. The aim of applying Young diagram is to inform the readers as briefly as possible about the beauty and conciseness of the mathematical rules underlying soliton equations. It is quite a surprise that soliton equations reduce to such simple Young diagram equations!This dissertation consists of eight chapters.In the first chapter, we introduce the historical background and the development of the soliton theory, and several commonly used methods for solving the nonlinear evolution equation. Then briefly describes the content and significance in this dissertation.In the second chapter, we give an indirect method of obtaining Wronskian solution of a general nonlinear evolution equation. Based on the method, we have constructed gen-eralized Wronskian solutions for the generalized Calogero-Bogoyavlenskii-Schiff(GCBS) equation and Boiti-Leon-Manna-Pempinelli(BLMP) equation.In the third chapter, we give a unique method to construction of a system of linear differential conditions.In the fourth chapter, based on the method of3.1.1, we have established gener-alized linear differential conditions of Wronskian solutions for the B-type Kadomtsev-Petviashvili(BKP) equation, isospectral B-type Kadomtsev-Petviashvili (IBKP) equation.In the fifth chapter, based on the method of3.1.1, we have established three generalized linear differential conditions of Wronskian solutions for the Kadomtsev-Petviashvili(KP) equation and six generalized linear differential conditions of Wronskian solutions for the (3+1)-dimensional Jimbo-Miwa(JM) equation. Moreover, we extend the linear partial differential condition and proved that KP and JM equation have extended Grammian solutions.In the sixth chapter, based on the method of3.1.2, we have established generalized linear differential conditions of Wronskian solutions for the Korteweg de Vries (KdV) equation, Modified KdV (MKdV) equation.In the seventh chapter, according to the properties of the computing of Young dia-gram, we discuss the relationship between the permutation group character and Young diagram expression coefficient.Finally, the summary of our work and the forecast for relevant work are prospected in the eighth chapter.