Applications Of Dressing Method To Soliton Equation

In this dissertation,firstly we introduce the different forms of dressing method,i.e.Zak-harov-Shabat dressing method,Zakharov-Mikhailov method,Riemann-Hilbert dressing meth-od,(?)-dressing method.Formally speaking,they respectively start from GLM equation,Lax pair,Riemann-Hilbert problem,(?)-problem.However,within the dressing scheme they are consistent.Secondly we solve several 1+1 dimensional soliton equations by using the different forms of dressing method,on the one hand we study the higher-order solutions of soliton equa-tions by virtue of 9-dressing method.The higher-order soliton solutions may be understood as the certain limit of multi-soliton solutions,but in essence,in the viewpoint of the inverse scattering theory the higher-order soliton solutions correspond to the multiple poles.Here we respectively study Gerdjikov-Ivanov equation,Newell's long wave-short wave equations,the coupled Sasa-Satsuma equations,these equations are associated with 2×2 matrix spectral prob-lem,3×3 matrix spectral problem,5×5 matrix spectral problem respectively.We start from the corresponding matrix(?)-problem and regain the Lax pairs of these equations,the general expressions of solitons and higher-order soliton solutions are obtained;On the other hand we develop Zakharov-Mikhailov method for a general coupled derivative nonlinear Schrodinger equations and obtain 1-soliton and 2-soliton solutions.In order to find the general expression of soliton solutions,we build the Darboux transformation and general Darboux transforma-tion to solve these equations,the N-soliton,higher-order soliton,breather and rouge solutions are obtained;Finally we discuss the application of the dressing method in solving the soliton solutions of the peakon equations.