Soliton Solutions And Darboux Transformation Of The Supersymmetric Nonisospectral KdV Equation

With the widespread application of supersymmetric systems in physics,some classic integrable systems have been supersymmetrized and supersymmetric equations have in-creasingly become the focus of research by many scholars.This thesis mainly studies the supersymmetric nonisospectral KdV equation.First,the supersymmetric nonisospectral KdV equation is derived by the method of spectral parameter expansion and then the Hirota bilinear method,Wronskian technique and Darboux transformation are used to solve the supersymmetric nonisospectral KdV equation.First of all,hierarchies for the supersymmetric isospectral and nonisospectral KdV equations are derived out by the method of spectral parameter expansion.Specifically,nonisospectral equations are given when spectral parameter ? envolves with time.Besides,in the case of n=1,we obtain the Lax pairs of the supersymmetric isospectral and nonisospectral KdV equation respectively.Secondly,we investigate the supersymmetric nonisospectral KdV equation by Hi-rota's bilinear approach and present one-soliton,two-soliton,three-soliton and even n-soliton solutions of the supersymmetric nonisospectral KdV equation.Besides,on the basis of the bilinear form,we use Wronskian technique to give Wronskian determinant solutions for the supersymmetric nonisospectral KdV equation.Then,based on the Lax pair of the supersymmetric nonisospectral KdV equation,the one-fold,two-fold and three-fold Darboux transformations of the supersymmetric nonisospectral KdV equation are given.And the Darboux transformation of the super-symmetric nonisospectral KdV equation is expressed in the form of super determinant.The Darboux transformation is applied repeatedly to obtain the n-fold Darboux trans-formation.Finally,the Darboux transformation of the supersymmetric variable-coefficient KdV equation is studied,and the transformation is constructed from the Lax pair of the equation.