Application Of Darboux Transformation To Some Soliton Equations

The solvability of equations is not only the focal point of studing the nonlinear partial differential equations but also a hot spot in soliton theory. This article focuses on three kinds of integrable equations:with a non-uniform KdV equation, Hirota equation and generalized Schrodinger equation. For these three kinds of equations, we mainly take advantage the way of the Darboux transformation, which for solving nonlinear partial differential equations has its own distinctive features and convenient. But according to the type and characteristic of equations, we use the different ways to analyze to solving the equations. First of all, we consider to the specific form of the Darboux transformation and the N order Darboux matrix, and then make use of the trivial solution and the non-trivial solution of equations to obtain soliton solution, like-soliton solution, breather solution. We take the generalized Schrodinger equation as an example and to take its the limit to get the rogue wave solution on the basis of breather solution. The rouge wave solutions have many parameters, which can be changed to adjust to the experiments. Rouge wave has been caused widespread attention from scientists, its important feature is appearing in a short time and the. local area own a large amplitude, and without any signs. Which as one of the major culprits causing shipwrecks, so it is crucial to study the rouge wave.Chapter2. Derives coupled with a non-uniform KdV equation Darboux matrix based on the specific form of the AKNS system.Chapter3. Studies Darboux transform and its reduction of the coupled Hirota equation, then solves the general expression of N-soliton solutions and breather solutions.Chapter3. Focuses on the generalized Schrodinger equation and its Darboux trans-formation, soliton solutions, like-soliton solutions, breather solutions and rouge wave so-lutions. This method has its own advantages to solve Schrodinger type equation based on Darboux transformation, and selects the appropriate the spectral parameters and the characteristic function the to get solutions by use the obtained N-order Darboux matrix. The key lies in which we can according to the Darboux transformation and periodic seed solutions to obain the rouge wave solutions.