Application Of Riemann-Hilbert Approach In Nonlinear Soliton Equations

As the development of soliton theory,finding exact solutions of nonlinear soliton equations has always been one of the main contents of soliton theory research.Among the many methods,the Riemann-Hilbert approach is an effective method for solving nonlinear soliton equations.This paper mainly studies two aspects work:find the N-soliton solution of the coupled nonisospectral Gross-Pitaevskii(GP)equation and extend the Riemann-Hilbert method to the multi-component nonisospectral GP equation;apply the Riemann-Hilbert approach to the matrix modified Korteweg-de Vries(mKdV)equation,and find its N-soliton solution.In the first chapter,a brief description of the development history of soliton theory and the research status of Riemann-Hilbert approach.The second chapter is to analyze the Lax pair of the coupled nonisospectral GP equation,obtain the Jost solution and the analytical properties of the scattering data.Moreover,construct the Riemann-Hilbert problem of the coupled nonisospectral GP e-quation and reconstruct the potential.Then analyze the symmetric relationship between the potential matrix and the scattering data,and the trace formula of the nonisospectral GP equation is given.A kind of Riemann-Hilbert problem for the coupled nonisospectral GP system is derived and the N-soliton solutions for the equation under the conditions of irregularity and reflectionless case also are presented.Meanwhile,the multi-component coupled nonisospectral GP system is generalized.The third chapter is focus on seeking the N-soliton solutions for the matrix mKdV equation,which generalizes to the multi-component mKdV equations and the coupled mKdV equations.By utilizing the technique of Riemann-Hilbert and investigating the spectral problem of the Lax pair,a class of Riemann-Hilbert problem will be discussed and established.Finally,in the case of irregularity and reflectionless,we derive the N-soliton solutions for a system of the matrix mKdV equation.The fourth chapter is the summary and outlook.