Study Of Lump Solution And Rogue Wave Solution For Some Soliton Equations

Along with the development of the soliton theory,searching exact solutions of non-linear evolution equations has been one of the hot topics in nonlinear science.Recently,the study of lump solution and rogue wave solution have attracted a special attention.Especially constructing and solving high-order lump solution and high-order rogue wave solution have been a significant and difficult research field in the nonlinear partial differ-ential point of view.This dissertation mainly consists of three parts.The first one is to present fundamen-tal lump solution to(3+1)-dimensional BKP equation and generalized M-Lump solutions to standard KPI equation.The second one is to show the lump solution and rogue wave solution to(3+1)-dimensional variable-coefficient BKP equation.The third one is to d-educe general high-order rogue waves of a(3+1)-dimensional generalized shallow water equation which are given in terms of determinants.Concretely,in the first part,we first derive the bilinear form of(3+1)-dimensional BKP equation and standard KPI equation.Then with symbolic computation by Maple,we search for positive quadratic function solutions to both equations.At the same time,through utilizing bilinear Backlund and nonlinear superposition formula,we give the extended M-Lump solutions for KPI equation from a trivial seed solution in determinant form.In the second part,we mainly talk about the rational solution of(3+1)-dimensional variable-coefficient BKP equation.We first deduce a bilinear form contacting parameters of the equation through the Rational-logarithmic transformation,then the lump solution and rogue wave solution are obtained by making use of the method of constructing for polynomial functions and a simple symbolic computation approach.Third part is mainly focus on the fundamental rogue wave and high-order,multi-rogue wave solutions of a(3+1)-dimensional generalized shallow water equation.We obtain the general high-order rogue waves of the equation,which are expressed in term of determinants based on the Hirota's bilinear method and KP hierarchy reduction method.