Research On Integratability And Analytical Solution Of Nonlinear Schrodinger Equation

As an important type of nonlinear evolution equation,the Schr?dinger equation is widely used in nonlinear quantum field theory,condensed matter physics,plasma physics,nonlinear optics,perturbation and phase transition theory,biophysics and many other fields.In actual research,the Schr?dinger equation is often extended to include various physical effects such as variable coefficients,complex coefficients,high latitude,high order,and non-locality.So far,the Schr?dinger equations have permeated almost all branches of physics.For example:Bose-Einstein condensate,superconductivity and superfluidity,ferroelectricity,ferromagnetism and antiferromagnetism in condensed matter physics;optical soliton communication,soliton laser and optical waveguide in nonlinear optics,etc.Therefore,it is of great significance to study the nonlinear Schr?dinger equation.Based on the classical method in soliton theory—Darboux transform and generalized Darboux transform method,this paper discusses their application and improvement in the process of solving two nonlinear Schr?dinger equations with variable coefficients,so as to obtain exact solutions with abundant equations.The structure of the full text is as follows:1.In the first chapter,we introduce the development history and research status of nonlinear evolution equations,soliton theory and strange wave,and briefly introduces several research methods commonly used in soliton theory:Painleve analysis method,Darboux transformation method,Hirota bilinear method and generalized Darboux transform method.At the end of this chapter,we list the main research work and the overall structure of the article.2.In the second chapter,we investigate a nonlinear Schr?dinger equation with variable coefficients.Based on the AKNS system of the nonlinear Schr?dinger equation,we first construct the Darboux transformation corresponding to the equation,and then use the series expansion technique to construct the corresponding Generalized Darboux transform.At the same time,based on the constructed generalized Darboux transform,various forms of solutions of the equations are obtained,and appropriate coefficient parameters are selected to make images of various solutions,showing the propagation paths of various solutions.By observing the image of the solution and analyzing the structural characteristics of the solution,we find that when different coefficient parameters are selected,the amplitude and velocity of the wave show certain regular changes,which helps us better understand the characteristics of this equation system and the corresponding solutions.Physical structure and propagation characteristics.3.In the third chapter,we apply the generalized Darboux transform to study a high-order nonlinear Schr?dinger equation with variable coefficients.Based on the Lax pair of this equation,we construct the Darboux transform corresponding to the high-order nonlinear Schr?dinger equation.Using the generalized Darboux transform theory,we construct the high-order strange-wave solution of the high-order nonlinear Schr?dinger equation.Similarly,we use the Maple drawing tool to make the trajectory image of the strange wave solution,and analyze the understood structural features.By observing the image characteristics of the strange wave solution,it is found that the strange wave can form a huge peak in a short period of time,with great energy and impact.The study of strange waves has very important research significance for many fields such as optics and oceanography.