Research On Exact Solutions Of Several Nonlocal Nonlinear Equations

Recently,nonlocal problems in integrable equations has attracted a lot of attention in soliton theory and solving the exact solution of nonlocal nonlinear equation is a significant project.In this paper,we’ll take the nonlocal Fokas-Lenells equation(nFL)and the coupled nonlocal Fokas-Lenells equation(cnFL)for example to explore the similarities and differences between local and nonlocal equations.The starting chapter describes the development history of soliton theory and the research status of nonlocal problems.Meanwhile,the method of Darboux transforma--tion and generalized Darboux transformation are introduced briefly.In second chapter,we construct the several exact solutions like bright/dark soli-tons,double periodic solutions and kink solutions of nonlocal Fokas-Lenells equation via Darboux transformation.First,using the method of undetermined coefficients,the DT of this equation was obtained.Choosing different seed solutions into iterative formula,we derived exact solutions of nFL.The discussion and comparisons between local and nonlocal FL equations are given at the end of the chapter.Chapter 3 study the rational solutions of the coupled nonlocal Fokas-Lenells equa-tion.The cnFL equation was derived from the 3×3 spectrum problem.We constructed generalized Darboux transformation of this equation and obtained the rational solu-tions and high-order rational solutions.These solutions can be classified into rouge waves and quasi-soliton solutions by different spectral parameters.It should be men-tioned that the solutions of nonlocal equation not only have the classical solution but also possess new solutions differ from the classic equation.