The Initial-Boundary Value Problem For Integrable Equation Based On The Riemann-Hilbert Approach

Riemann-Hilbert approach,as an important method and technique to study nonlinear equations,has been studied by many researchers in recent years.This paper mainly studies the prolongation structure method and Riemann-Hilbert approach in the application of nonlinear integrable equation.For a specific nonlinear integrable equation,based on the prolongation structure method and some related properties of Lie algebra,we can get the Lax pair of the equation,and then we can prove that the solution of the initial-boundary value problem for the equation can be represented by the solution of the Riemann-Hilbert problem.The specific arrangement of this paper is as follows:In the first chapter,we mainly introduce the research background and current situation of soliton theory,especially some important techniques to obtain the exact solutions of the soliton equations.And then we describe the development and application of the prolongation structure method and the Riemann-Hilbert approach at home and abroad in detail.Besides,we analyse the research background of the mNLS equation.Finally,the main work and structural arrangement of this paper are explained.In the second chapter,the prolongation structure method is described detailed.For a specific nonlinear partial differential equation,the basic steps to get the Lax pair of the specific equation by using the prolongation structure approach are given.And then we analyse the prolongation structure of mNLS equation via this approach and some related knowledge of Lie algebra.Finally,the Lax pair of the mNLS equation is obtained,In the third chapter,the initial-boundary value problem for mNLS equation on the half-line is analyzed.Assuming that the potential function at infinity decay sufficiently,we reconstruct the Lax pair to obtain the corresponding matrix Riemann-Hilbert problem.The specific transformation is introduced to obtain the matrix differential form of the Lax pair.Besides,the spectral function is defined according to the initial-boundary value.In addition,the key to solve the Riemann-Hilbert problem is to find specific jump conditions,the certain jump and residue relationships are investigated in the following.Furthermore,the potential function can be represented by the solution of this Riemann-Hilbert problem.