Darboux Transformation And Exact Solutions Of Schwarzian KdV Equation And Derivative Manakov Equation

In this paper,we mainly study the Darboux transformation of two important soliton equations and the corresponding exact solutions,that is the Schwarzian KdV equation and the derivative Manakov equation.This paper is divided into five parts.In the first part,it is the introduction,which mainly introduces the development of soliton theory.In the second part,we first consider the spectral problem of two(1+1)dimension Schwarzian KdV equations,and obtain the(2+1)dimension Schwarzian KdV equation,and then we construct the N-fold Darboux transformation of the corresponding spectral problem in two cases.In the third part,we select the seed solution and use the Darboux transformation to obtain the N-soliton solution of three Schwarzian KdV equations.In particular,we analyze the asymptotic property of the exact solutions of the(2+1)dimension Schwarzian KdV equation when N=1,2,3,4 and select the appropriate parameters to make the corresponding graphs.In the fourth part,we first introduce a derivative Manakov equation related to the 4×4 spectrum problem,and then construct a Darboux transformation of its spectrum problem and derivative Manakov equation.In the fifth part,we select the seed solutions and use the Darboux transformation to obtain the exact solutions of derivative Manakov equation when N=1,and then select the appropriate parameters to make the corresponding graphs.