Apply Darboux Transformation And Hirota Bilinear To Solve Explicit Solutions Of Some Soliton Equations

Soliton theory is one of the mainstream direction of nonlinear scientific research,and solving the exact solution of nonlinear soliton equations is a basic and important subject.All kinds of exact solutions are masterfully constructed by mathematical method,then both graphs are analysed to explain natural phenomena scientifically and describe the perceptual world,which has certain practical significance.This article separately takes Darboux transformation and Hirota bilinear method as the research methods to solve a few kinds of exact solutions of generalized coupled nonlinear Fokas-Lenell equation and(2+1)multi-component Marcarri equation,including soliton,breather and rogue wave solutions.Among them,the rogue wave solution is a hot topic to study soliton theory content in recent years,also is about the chief research content.The starting chapter briefly summarizes the development of soliton theory and rel-evant research methods about solving the nonlinear partial differential equations,espe-cially introduced some examples of solving exact solutions by Darboux transformation and Hirota bilinear method.Finally,we took a further discussion about the produce and research status of the rogue wave to prepare for the following work.The soliton,breather,and rogue wave solutions of the original equation are derived on the basis of the Darboux matrix of the generalized coupled Fokas-Lenells system in second chapter.Firstly,using Darboux matrix,the iterative formula of solution was derived,from a basic solution of equation to the new one.Then we set the seeds of equation to solving out to the eigenfunction of the Lax pair.The soliton and breather solutions can be obtained from the iteration formula.If taking a limiting process,the rogue wave solution can be obtained.The third chapter studyed the(2+1)-dimensional multi-component Marccari system and the rogue wave solutions of it.The original equation is transformed into equivalent equation containing new variables by variable substitution on the basis of the Hirota bilinear equation.The rational solutions are obtained by using the ? function.Then the rogue wave solutions of the equation can be derived.In the paper,the fundamental rogue wave and the second order rogue wave are made detailed analysis of the dynamics.