The Initial-boundary Value Problem And Darboux Transformation For Integrable Systems

In this PhD dissertation,we mainly discuss applications in integrable systems of the Riemann-Hilbert problem and Darboux transformation.We use the Fokas unified method to analyze the initial-boundary value for the Chen-Lee-Liu equation and the complex Sharma-Tasso-Olver equation on the half line.Furthermore,we show that the solution can be expressed in terms of the solution of Riemann-Hilbert problem.Starting from two new discrete spectral problems,we derive two hierarchies of discrete integrable systems,respectively.It is shown that these two hierarchies are integrable in Liouville sense and possess Hamiltonian structures.The Darboux transformations for the two hierarchies are further established with the help of gauge transformations of their Lax pairs.As an application,some new exact solutions for the two hierarchies of discrete integrable equations are obtained.The dissertation is arranged as follows.In chapter 1,we mainly introduce the background and developments of the Riemann-Hilbert problem,Darboux transformation,Wu Wenjun's mechanical thought of mathematics,and symbolic calculation method.We also present main research contents and results in this dissertation.In chapter 2,we apply Fokas unified method to analyze the initial-boundary value for the Chen-Lee-Liu equation on the half line(-?,0]with decaying initial value.It is shown that the solution can be expressed in terms of the solution of Riemann-Hilbert problem.The relevant jump matrices are explicitly given in terms of the spectral functions which in turn are defined in terms of the initial values and boundary values.In chapter 3,we apply Fokas unified method to analyze the complex Sharma-Tasso-Olver equation on the half line[0,?).It is shown that the solution of the initial-boundary problem can be expressed in terms of the solution of a Riemann-Hilbert problem.Comparing with the Chen-Lee-Liu equation in the chapter 2,the order of the complex Sharma-Tasso-Olver equation is much higher,so the analysis and calculations of the complex Sharma-Tasso-Olver equation become more complicated.In chapter 4,starting from a new discrete isospectral problem,a hierarchy of integrable lattice models are derived.It is shown that the hierarchy is integrable in Liouville sense.Fur-ther,a Darboux transformation of the hierarchy is established,from which the exact solutions are further obtained.In chapter 5,starting from a new discrete spectral problem,we derive two hierarchies of discrete integrable systems.It is shown that the discrete hierarchies are all integrable in Liou-ville sense and posses Hamiltonian structures.Furthermore,their N-fold Darboux transfor-mations are established with the help of gauge transformations of Lax pairs.As an application,we present the explicit forms of the exact solutions for the two discrete integrable equations in the hierarchies in the case of = 1 and N = 2.