The Analysis And Research On The Rogue Wave Solutions Of The Couple Nonlinear Schr(?)dinger Equations

Rogue wave is a kind of marine disaster with a great degree of harm.It is a very special wave which exists in a short time and has high amplitude. It is generally believed that rogue wave is caused by modulation instability, but not all the modulation instability can produce rogue waves.The accuracy of schrodinger equation can only be tested by experiment, which is a hypothesis of quantum mechanics, and is combined with the idea of material wave and wave equation. It can be used to describe the motion of microscopic particles. Not only in fluid mechanics, plasma physics and Bose Einstein condensation fields, the schrodinger equation has been also widely used in nonlinear Schr(?)dinger equation which can also be used to describe the nonlinear dynamic characteristics of optical and Heisenberg ferromagnetic spin chain problems, this is also the focus of research.In this paper, we use the generalized Darboux transformation to analyze the two kinds of high order nonlinear schrodinger equations.Darboux transformation is a iterative process of solving, it is closely related to the standard transformation, and it is a special kind of standard transformation. standard transformation can be summarized as that an equation equation can keep the deformation in a group of transformation,the transformation is called the standard transformation. Darboux transformation cannot be directly used to obtain higher-order rogue-wave solutions, so we need use limit knowledge to improve it in order to get higher-order rogue-wave solutions. In the promotion of using Darboux transformation, we need obtain the solutions of the Lax pairs corresponding to the nonlinear equations. Lax pairs is a set of nonlinear partial differential equations, the solution should be accomplished using mathematical software or similar coefficient method. In this paper, we firstly transform the original Lax pair into a linear partial differential equation group through a matrix transform, then the solutions of Lax pairs can be obtained by simple algebra eigenvalues and eigenvectors.The method of transforming Lax pair into linear partial differential equations by matrix transformation can be applied not only in(1+1)-dimensional nonlinear schrodinger models but also in(2+1)-dimensional nonlinear schrodinger model.In this paper, we explore the causes and occurring mechanisms of rogue waves through analyzing the rogue waves and the modulation instability of the (1+1)-dimensional nonlinear schrodinger models in a multi-mode fibre and a (2+1)-dimensional fourth-order nonlinear schrodinger equation in a Heisenberg ferromagnetic spin chain.