Peakon Wave And Rogue Wave Solutions Of Nonlinear Evolution Equations

With the rapid development of natural science and social sciences,people pay more and more attention to the study of nonlinear problems,especially the problems of nonlinear mathematical physics are widely concerned by many scholars in mathematics and physics.The theory of soliton and integrable system is universally applied to physics such as superconductivity,fiber optic communication,plasma physics,fluid mechanics,quantum field theory and biology,chemistry,finance,oceanography,etc.Therefore,the study of this theory has always been one of the most active fields.The explicit solution of nonlinear evolution equation is an important aspect of soliton and integrable system theory.This paper mainly studies the peakon wave solution,rogue wave solution and exact solution of several nonlinear evolution equations.The first chapter,as an introduction,the research background,research status and significance of several nonlinear evolution equations are introduced.The second chapter,we study a Camassa-Holm(CH)equation with n-component and a two-component CH-type equation with parameter b based on the distribution theory.According to the conditions satisfied by the parameters,the double peakon solutions of the two equations are discussed in detail,and some properties of the solutions are obtained by graphic analysis.The third chapter,two water wave models,namely(1+1)-dimensional Benjamin-Ono(BO)equation and(3+1)-dimensional BKP equation,are studied.A symbolic computation approach based on bilinear equation is established to solve the higher order rogue wave solutions of BO equation and BKP equation.In addition,some real parameters of multi-soliton solutions are appropriately extended to complex parameters.When they are taken asconjugate complex numbers in pairs,the multiple breath solutions of BO equation are obtained.The fourth chapter,we get the exact solutions of the higher order nonlinear Schr(?)dinger equation by using the auxiliary equation method.Finally,a simple summary and some further prospects are proposed.