Integral Systems And Random Matrices:a Riemann-Hilbert Approach

In this PhD thesis, we analyse the Riemann-Hilbert problem and integral system, or-thogonal polynomials, random matrices. Using the Riemann-Hilbert method or nonlinear steepest descent method to analyse the long-time asymptotics behavior of the solution of the nonlinear evolution equations; using Fokas method to analyse the initial-boundary value problem of the high order nonlinear evolution equations; using Riemann-Hilbert problem and Deift-Zhou method to analyse the university problem of a special type of random matrix model. According to the orders of the Riemann-Hilbert problem, we split the analysis into three parts:In chapter 2, we consider the long-time asymptotics behavior of the solution of the Fokas-Lenells equation with decaying initial value. Because th eLax pair of the Fokas-Lenells equation is 2×2, the Riemann-Hilbert problem of this equation is also 2×2. Based on the nonlinear steepest descent method (or Deift-Zhou method), according to a series of transformations of the jump contours, we can change the Riemann-Hilbert problem into a model Riemann-Hilbert problem. And we make some changes on this model Riemann-Hilbert problem, it can be written as a parabolic-cylindrical equation. Using the asymptotics behavior of this parabolic-cylindrical equation, we can finally find the asymptotics behavior of the solution of the Fokas-Lenells equation.In chapter 3, we consider the 3×3 spectral problem of the Sasa-Satsuma equation and three-wave equation. We can formulate a 3×3 Riemann-Hilbert problem to the initial-boundary value problem on the half-line of these two equations. But as the Lax pair of the Sasa-Satsuma equation has derivative term, we need know these derivative boundary data. However, for a well-posed problem, the known data is too much. And these too much boundary data is not independent, they satisfy a so called global-relation. And the most important problem is to determine these too much data based on the initial data and boundary data. We get the nonlinear boundarize condition to the Sasa-Satsuma equation. However, there doesn’t exist too much data for the three-wave equation. The analysis of the three-wave equation becomes simpler. Based on the Riemann-Hilbert problem of the three-wave equation, we get the existence and uniquely conditions of this Riemann-Hilbert problem.In chapter 4, we consider the random matrix model with external source using Riemann-Hilbert problem. When the external source matrix has three distinct eigen-values, we analyse the large n-limit of the correlation function of this model. Because of three distinct eigenvalues, the Riemann-Hilbert problem is 4 x 4. We can simplify the Riemann surface of this model using the symmetry of the three eigenvalues. Assuming that the Riemann surface has six distinct real branch points, we can get the asymptoitcs of the correlation functions as n goes to infinite.