Construction Of The New Supersymmetric Systems And Integrability Of The Supersymmetric Integrable Systems

In this thesis,using Hirota's method,we mainly discuss how to seek new supersymmetric integrable systems.Moreover,we study the integrability of the resulted supersymmetric integrable systems.In details,1.In 1980,Nakamura and Hirota found so called the second modified KdV equation from the bilinear BT for the modified Korteweg-de Vries(MKdV) equation.In this thesis,we successfully apply the method to supersymmetric cases and obtain a supersymmetric second modified KdV equation(ssMKdV).Moreover,we present one-soliton, two-soliton and three-soliton solutions of the ssMKdV equation.2.We will construct a new supersymmetric Classical Boussinesq system from Hirota bilinear method.This new system is different from the version of Brunelli and Das.Then,we perform a Painlevéanalysis for the new equations and show that the equations pass the Painlevétest.Finaly,we calculate one-soliton,two-soliton solutions of the new equations.3.Finaly,we study the N=2 supersymmetric Korteweg-de Vries(SKdV) equations within the framework of the Hirota bilinear method,we succeed in obtaining their bilinear forms.We also construct the solutions for both equations and find a simple B(?)cklund transformation and Lax representation for the SKdV₁ equation.In addition, we construct the bilinear form for the t₄ flow of the supersymmetric Classical Boussinesq hierarchy.