Study On Solving Soliton Of The Schrodinger Equation With Variable Coefficients

In recent years,the study of the Schr(?)dinger equation with variable coefficients has been paid more and more attention by researchers.It can be seen in nonlinear optics,quantum fields,and so on.Soliton solution is an important part of nonlinear equations.Three equations are considered:the Schr(?)dinger equation with variable coefficients in the optical fiber,the equation in physical optics,and the Schr(?)dinger equation with variable coefficients and disturbance item,giving equation source and application fields,mainly from the bilinear method and the Painlev(?) analysis method.For the first kind of equation,the appropriate variable transformation is used to obtain the bilinear equations.Then the solution to the bilinear equations are obtained by the expansion truncation.Finally,the single soliton solution,the double soliton solution and the n-order soliton solution to the original equation are obtained.For the second equation,combining WTC algorithm,we introduce the series transformation and analyze the first term and the resonance point to obtain the solution to the equation.For the third equation,we explore the inconvenience for bilinear method,give analysis of resonance conditions and derivation of B(?)cklund transformation,and then give five solutions to the equation with disturbance term and the corresponding analysis,the article is organized as follows:In the first chapter,we introduce the development history of soliton research,bilinear methods,Painlev(?) analysis and research background.The second chapter is about the bilinear method,based on the Schr(?)dinger equation with variable coefficients.It introduces the basic idea,operator properties,common transformations,the solution process of bilinear equations,and solves several kinds of soliton solutions to the original equation:single soliton solution,double soliton solution,and n-order soliton solution.The third chapter is about the Painlev(?) analysis.It introduces the basic theory,analyzes the process,and obtains the necessary conditions for the factor of the Painlev(?) analysis,analyzes several resonance points,and gives an example.In the fourth chapter,we discuss the Schr(?)dinger equation with variable coefficients and disturbance item,analyze the inapplicability of the bilinear method,and use the Painlev(?) analysis of the classification to obtain the B(?)cklund transformation and state conditions of the equations.Finally,we obtain several soliton solutions to the equation:kink-type soliton solution,bell-type soliton solution,and analyze the solution.In the fifth chapter,we summarize the basic process of two methods,the analysis of the disturbance item scheme and the discussion of future research.