Application Of Wronskian Technique And Pfaffian Technique To Some Soliton Equations

This dissertation mainly consists of three parts. The first one is to present double Wronskian solution to nonautonomous nonlinear Schrodinger equation(NNLS). The second one is to deduce soliton solutions to a modi-fied KdV equation with non-uniformity terms, what's more, we also present double Wronskian solution and give the proof. The Third one is mainly about the soliton solution expressed by Pfaffian representation of a cou-pled modified KdV equations with non-uniformity terms.Concretely, in the first part, we first derive the bilinear form of NNLS in the bright case. Then we give the double Wronskian solution to the NNLS based on Wronskian technique. At the same time, another kind of exact solution named rational-like solution, which are represented in the term of double Wronskian, are presented for the dark case.In the second part, we presented two kinds of exact solution to a modi-fied KdV equation with non-uniformity terms. we first deduce a new bilin-ear form of the equation through the rational-logarithmic transformation. Then the N-soliton solution are presented and a generalized double Wron-skian solution are constructed. Finally, we give the proof.The third part is mainly focus on the multisoliton solution of a cou-pled modified KdV equations with non-uniformity terms. We first give the single soltion solution and the two-soltion solution based on the Hirota bi-linear method. However, it is difficult to deduce explicit expression of the multi-solition solutions following the same procedure. With the elegant identities of Pfaffian, we give the representation of the multi-solition solu- tion in the form of Pfaffian. Moreover, we give the proof.