Riemann-Hilbert Problem Of The Derivative Nonlinear Schr(?)dinger Equation With Vanished Boundary Conditions And Its Analytical Solution And Integrability

Nonlinear science involves a wide range of disciplines.Soliton theory is a very important branch of nonlinear science,which has wide applications in physics,chemistry and engineering,and promotes the rapid development of Lie groups,geometric manifolds,function theory,condensate physics,nonlinear optics and fluid mechanics.In this thesis,the soliton solution of the derivative nonlinear Schrodinger(DNLS)equation is studied by Riemann-Hilbert method.We propose a modified RiemannHilbert problem and inverse problem of the DNLS equation are given with the zero boundary condition.Based on this inverse problem,the explicit formulae for the N-th order soliton solution and N-th order positon solution of the DNLS equation are obtained.which correspond to N pairs simple poles and one pair of N-th order poles of the Riemann-Hilbert problem.By applying the Cauchy-Binet formula,the N-th order solitons are given explicitly.Additionally,the explicit first-order soliton and second-order positon are displayed as examples.The main structure of this thesis is as follows:The first chapter mainly introduces the origin and progress of soliton theory,and briefly introduces some traditional methods such as scattering inversion,Darboux transform,Backlund transform,Hirota bilinear method and Riemann-Hilbert method in solving nonlinear differential equations.In chapter 2,a revised Riemann-Hilbert problem and an inverse problem of the DNLS equation with zero boundary condition are studied.Based on this inverse problem,the explicit formulae of the N-th order soliton solution and bound-state soliton solutions of the DNLS equation are obtained.The third chapter briefly summarizes and discusses the content of this thesis.