The Mixed Soliton Solution And Rogue Wave Solution Of The Nonlinear Derivative Schrodinger Equation

Nonlinear partial derivative equations(NPDE)originate mostly in physics,chemistry,life science and other scientific fields.And they are abstract physical models,which have distinct physical meanings.There are two systems for NPDE,one is integrable(weakly nonintegrable),and another is of nonintegrability.However,the research on integrable NPDE is a hot topic for scientists because it can generate soliton solutions.Derivative nonlinear Schrodinger(DNLS)equation is one of the strictly integrable NPDEs,which is the mathematical model of Alfven waves in plasma.However,it can also describe the sub-picosecond or femtosecond pulses in single-mode optical fibers,the weak nonlinear electromagnetic waves in(anti-)ferromagnetic or dielectric systems under external magnetic fields.There are three prevailing methods,inverse scattering transform(IST),Darboux transform(DT)and Hirota bilinear method,which can solve DNLS equation.Among them,the IST is used to solve DNLS equation under constant boundary conditions mostly.The other two methods can solve DNLS equation under non-constant boundary condition conveniently.This thesis focuses on solving DNLS equation under nonvanishing constant boundary condition and plane wave background.The first part of this thesis deals with the research background of NPDE and soliton theory.Several commonly used NPDE's solving methods are introduced in the second partThe fourth part considers solving DNLS equation under nonvanishing constant boundary condition,and constructing a mixed solution which combines with pure solitons and breathers via IST.The Hirota method is used to construct space periodic solution of DNLS equation under a plane wave background in part five.Further,the long wave limit of the first order space periodic solution is taken to deduce the rogue wave solution.Conclusions and outlooks are given in part six.