Study On The Integrability And Exact Solutions Of Some Nonlinear Evolution Equations With Variable Coefficients

With the development and wide application of nonlinear science,the research on nonlinear evolution equation has gradually become a frontier topic.Since the nonlinear models with constant coefficients are generally established under ideal conditions,it can not fully describe the actual physical phenomena.Therefore,in view of the inhomogeneity of the transmission medium and boundary conditions,the variable coefficient models established under the influence of external environment are more practical than the constant coefficient ones.Consequently,the research on nonlinear systems with variable coefficients has turned into a research hotspot of nonlinear science.In this paper,based on the classical methods in soliton theory:Darboux transformation and Hirota bilinear method,the application and improvement of these methods in solving two nonlinear evolution equations with variable coefficients are discussed,so as to obtain rich and exact solutions of the equations.The structure of the full text is as follows:In the first chapter,we first introduce the development history and research status of nonlinear evolution equations and soliton theory,and four research methods commonly used in soliton theory are also introduced:Darboux transformation,Backlund transformation,Painleve analysis and Hirota bilinear method.Finally,the main work and structure of this paper are given.In the second chapter,on the basis of the Hirota bilinear method and symbolic calculation,a(2+1)dimensional nonlinear coupled partial differential combination equation with variable coefficients is studied.Through the selection of coefficients,the equation can be reduced to three kinds of known equations,the breaking soliton equation,the shallow water wave equation and pKP equation with variable coefficients.Through the bilinear form of the equation,combined with the improved three wave method and the positive quadratic function method,the N-soliton solutions,breather wave solutions and Lump solutions of the equation are obtained respectively,and the appropriate coefficient parameters are chosen to show the propagation trajectories of various solutions.Through analysis,when taking different coefficients,such as constants,polynomials,exponential functions,trigonometric functions,etc.,we find that the amplitude,shape and velocity of the wave change regularly,which makes us better understand the physical structure and propagation characteristics of the solutions of the equation.In the third chapter,the exact solution of a higher-order dispersion nonlinear Schrodinger equation under the influence of variable coefficients is deeply studied.Firstly,we analyze the classical Darboux transformation and the generalized Darboux transformation of the equation,and use the classical Darboux transformation to obtain the one-soliton,the two-soliton and the first-order breather solution,and analyze the propagation characteristics under the conditions of different coefficients.With the increase of the number of iterations,the calculation difficulty of the classical Darboux transformation increases rapidly.Therefore,in order to simplify the calculation,we use the generalized Darboux transformation theory to construct the first-order rogue wave solution and the second-order rogue wave solution,analyze the structural characteristics of the solutions,and show that the rogue wave can form a peak in a short time and has great energy and impact,which is of great research significance for the marine environment.The last chapter is the summary of this paper and the prospect of future research work.