Applications Of Fredholm Integral Equations In Nonlinear Integrable Models

There are two parts in this dissertation: in the first part,a new approach to the initial-boundary value problem of nonlinear integrable equations is constructed by using Fredholm integral equations;in the second part,based on Fredholm integral equations and Darboux transformations,exact solutions to some nonlinear integrable equations are obtained.Though important,the initial-boundary value problems of nonlinear integrable equations are difficult.Therefore,the study on initial-boundary value problems is about transforming differential equations into other problems,like Riemann–Hilbert problems,so that the original problem is solved as long as the latter one is solved,exactly or numerically.Because matrix Riemann–Hilbert problems are still open,we focused our eyes on transforming initial-boundary value problems into Fredholm integral equations,which is better studied in many aspects.In Chapters 2 to 4,by using Fredholm integral equations,we shall inspect the initial-boundary value problem of the nonlinear Schr?dinger equation on a finiteinterval,the initial-boundary value problem of the modified Korteweg-de Vries equation on a half-line,and the initial-boundary value problem of the matrix nonlinear Schr?dinger equation on a half-line.It takes several steps to transform initial-boundary data into Fredholm integral equations.The first step is to determine the scattering data,the jump matrix and the residue relations by using the inverse scattering transform and the Fokas unified transform.The jump matrix in Fokas unified transform method in analogy of the continuous data in the inverse transform method,and the residue conditions,the discrete data.To the best of our knowledge,the jump matrix and the residue relations are not independent.This is because the entries that determine the jump matrix is analytic in a half-plane,while the continuous scattering data is only defined on the real-line.This property is taken into consideration in our Fredholm integral equation method.The second step is to derive the relation relating the scattering data and the eigenfunctions by means of Fredholm integral equations.Observing the components of the eigenfunctions involve an invertible Volterra operator,we simplify the relations,arriving at a Fredholm integral equation.The unknown functions in the Fredholm integral equation are as many as the potentials of the corresponding nonlinear equation.Though it may involve improper integrals,we managed to show that the Fredholm integral equation is uniquely solvable.The third step is to find a condition,known as an admissible condition,for the solvability of the problem.This is no boundary data in the problems solved by inverse scattering transform method.So there is no such conditions therein.In the Fokas unified transform method,a necessary condtion for the solvability is given by a so-called global relation.We managed to present a conditon that is both necessary and sufficient.In Chapter 5,by constructing a Fredholm with separatable kernel,we managed to present exact solutions for the matrix nonlinear Schr?dinger equation and the matrix modified Korteweg–de Vries equation.This is an application of the Fredholm integral equations obtained in previous chapters.In Chapter 6,we shall construct a Darboux transformation for the two-component Drifeld–Sokolov–Satsuma–Hirota equation,and then solve a generalized Harry Dym equation by means of the traveling-wave method.