Research On Soliton Solutions Of Several Multi-component Integrable Equations

The exact solution of the equations has always been the focus and even the difficulty in the study of nonlinear partial differential equations,especially the solution of the coupled multi-component integrable equations is a research hot topic.This paper mainly studies the two types of integrable equation:Mel'nikov system and Maccari system According to the common characteristics of these two kinds of multi-component equations,the Hirota bilinear method and the KP reduction technique are used to find the soliton solution,and the bound state solution is further obtained.As the important method to derive the soliton solutions of multi-component integrable equations,compared with the Hirota bilinear method,the KP reduction technique is more elegant in form and is often used to construct the soliton solution,rational solution,the rogue wave solution,etc.First of all,from the bilinear form of the system,we can find bilinear equations in a similar form in the KP hierarchy;Next,through variable substitution and reduction,the bilinear equation in the KP hierarchy can be transformed into the bilinear form of the equation to be solved.Finally,the soliton solution in Gram determinant can be obtainedChapter 2.We derive the bright-dark mixed multi soliton solutions for the coupled Mel'nikov system with the help of the KP hierarchy reduction method.In addition,the dynamics of one and two solitons are also discussed.The dynamical analysis shows that the collision of the mixed two solitons is elastic.Furthermore,the mixed soliton bound states are also investigatedChapter 3.Taking the coupled Maccari system for instance,by virtue of Hirota bilinear method and the KP hierarchy reduction method,the N-dark soliton solutions are further obtained in terms of determinants.In addition,in contrast with bright-bright soliton collisions,the dynamical analysis shows that the collisions of dark-dark solitons are elastic and there is no energy exchange among solitons in different components.What's more,we investigate the dark-dark soliton bound states including stationary and moving ones.Besides,we complete the generalization from two-component to M-component,and the general N-dark soliton solutions of the multi-component Maccari system are also constructed.