Applications Of Riemann-Hilbert Approach To Nonlinear Evolution Equations

The thesis mainly study some nonlinear integrable models which are associ-ated with 2×2 matrix spectral problems by the Riemann-Hilbert approach.On the basis of the spectral analysis for the Lax pair and related research for the scattering datas,a Riemann-Hilbert problem of the nonlinear integrable mod-el is established.Using the inverse scattering transformation and solving the Riemann-Hilbert problem of the nonlinear integralbe model,N-soliton solutions of the nonlinear integrable model are obtained.In Chapter 2,the Gerdjikov-Ivanov equation is studied.First,based on the spectral analysis for the Lax pair,we can use the Riemann-Hilbert approach to solve the Gerdjikov-Ivanov equation.Then,the trace formula and the solution formula of the Gerdjikov-Ivanov equation are given.Last,N-soliton solutions of the Gerdjikov-Ivanov equation are obtained by solving Riemann-Hilbert problem for the reflectionless cases.In Chapter 3,the negative power flow equation of the Kaup-Newell model is investigated.Different from that in Chapter 2,we introduce a gauge transfor-mation for the potential matrix to meet some certain symmetrical relationships.Then we obtained N-soliton solutions for the negative power flow equation of the Kaup-Newell model via Riemann-Hilbert approach.Similar to that in Chapter 3,the degenerate equation of the Manakov model is studied in Chapter 4;the combined nonlinear Schrodinger and Gerdjikov-Ivanov equation is studied in Chapter 5.On the basis of the gauge transformation and the spectral analysis for the Jost solutions,the Riemann-Hilbert problem is es-tablished and N-soliton solutions are obtained.In Chapter 6,a new nonlinear wave equation is studied.First,according to the spectral analysis for the Lax pair,the Riemann-Hilbert problem of the nonlinear wave equation is established.Then,we obtain the N-soliton solutions of the nonlinear wave equation by solving Riemann-Hilbert problem.