Applications Of Bilinear Approach In Some Integrable Systems

Blinear approach plays an important role in the research of the integrable theory.In this thesis,by means of bilinear approach,we mainly investigate a so-called extended homoclinic orbit solutions of the Kortewege-de Vries type bilinear equation,the Fokas-Lenells equation and the derivative nonlinear Schr¨odinger equation.The Kd V-type bilinear equations always allow 2-soliton solutions.In this paper,for a general Kd V-type bilinear equation,we interpret how the so-called extended homoclinic orbit solutions arise from a special case of its 2-soliton solution.Two properties of bilinear derivatives are developed to deal with bilinear equation deformations.A nonintegrable(3+1)-dimensional bilinear equation is employed as an example.Then,the Fokas-Lenells equations are investigated via bilinear approach.We bilinearize the unreduced Fokas-Lenells system,derive double Wronskian solutions,and then,by means of a reduction technique we obtain variety of solutions of the reduced equations.This enables us to have a full profile of solutions of the classical and nonlocal Fokas-Lenells equations.Some obtained solutions are illustrated based on asymptotic analysis.As a notable new result,we obtain solutions to the FokasLenells equation,which are related to real discrete eigenvalues and not reported before in the analytic approaches.These solutions behave like(multi-)periodic waves or solitary waves with algebraic decay.Finally,we investigate solutions of the coupled Kaup-Newell equations and their reductions by means of bilinear method and a reduction technique on double Wronskians.The reduced equations include the derivative nonlinear Schr¨odinger equation and its nonlocal version.Solutions of the reduced equations,including those related to real eigenvalues of the Kaup-Newell spectral problem,are presented,and their dynamics are illustrated.The research of the thesis develops more applications of the bilinear approach and provides insight into integrable equations associated to the Kaup-Newell spectral problem.