Painlevé?Analysis?and?Solutiotns For A Class Of Partial Differential Equations

With the continuous development of science and technology,linear science has long been unable to meet the actual needs,so non-linear problem research has become one of the hot topics of current scientific research.Non-linear problems are usually described by nonlinear partial differential equations,so whether to explore more equations and find more solutions of equations becomes the primary task of nonlinear scientific research.Based on symbolic computation,the Painlevé properties and analytic solutions of the variable coefficient nonlinear Schr?dinger(NLS)equation are investigated,which involves four arbitrary functions of space-time.Firstly,the relationship between the four variable coefficients are derived with WTC method when the equation is Painlevé integrable.Among the four variable coefficients of the equation,the first two are two-order dispersion of longitudinal distance and nonlinear coefficient respectively,and the last two are the real and imaginary parts of the fiber loss factor.Three special forms of rational function solutions are derived with Painlevé truncation method.Finally the partial solutions of the equation are obtained by using the variable separation method.The obtained results are the extension of the existing conclusions.The thesis is arranged as follows:The first chapter introduces the background,significance and the research status at home and abroad of the research topic.The second chapter introduces the source of the subject and the main research content of this paper.In the third chapter,based on symbolic computation,the Painlevé properties of the general variable coefficient nonlinear Schr?dinger(NLS)equation is investigated,and the relationship between the four variable coefficients are derived with WTC method when the equation is Painlevé integrable.In the fourth chapter,the partial analytic solutions of the equations are obtained by using the Painlevétruncation method and the variable separation method when the equation is Painlevé integrable.These solutions contain three special forms of rational function solutions and variable separation solutions.And two kinds of variable separation solutions for the variable coefficient Schrodinger equation are given.The fifth chapter gives conclusion and prospect.