A Research On Soliton Solutions And Quasi-periodic Solutions Of The Supersymmetric Variable Coefficient KdV Equation

The variable coefficient KdV(VCKdV)equation.as an important nonlinear evolution model in solit.on theory,has attracted great attention to mathematicians and physicists in recent years.In this paper,using direct method.we mainly discuss how to seek new supersymmetrie integrable systems and study new integrability of the resulted supersymmetric integrable systems.Moreover,we derive the solutions for the supersymmetric VCKdV equation by Hirota method,,Backlund transformation and super Riemann theta function.First,supersymmetric form of the VCKdV equation is given by direct method.The integrability of this model is examined through the Painleve analysis.According to variable transformation and bilinear method,the bilinear form of this equation is derived.Soliton solutions for supersymmetric VCKdV equation can be obtained by supersymmetrv bilinear derivative.Then starting from the bilinear form of supersymmetric VCKdV equationwe get the generalized bilinear Ba.cklund transformation.By the commutability of the bilinear Backlund transformation,we give the one-soliton solution,two-soliton solution and three-soliton solution for the supersymmetric VCKdV equation.Third,based on Hirota bilinear method and Riemann theta function,quasi-periodic solutions for the supersynmmetric VCKdV equation are studied.In addition,we give the asymptotic relations[between quasi-periodic wave solutions and soliton solutions.