Symbolic Computation On Some Nonlinear Continuous And Discrete Equations

Nonlinear evolution equations are used to describe the nonlinear phenomena in different fields,such as optics,fluid mechanics and condensated matter physics.We discuss the rogue waves,solitons,lump waves and periodic waves for some continuous and discrete nonlinear evolution equations.The main contents of this dissertation are as follows:In chapter 1.we introduce the nonlinear science and the research progress of the soliton,rogue-wave and lump solutions.We also introduce the mathematical methods in this dissertation.The main work and the organization of this dissertation are presented.In chapter 2,we study a(2+1)dimensional variable coefficient Gross-Pitaevskii equation.Using the Hirota method and Kadomtsev-Petviashvili(KP)hierarchy reduc-tion,dark solitons and rogue-wave solutions are obtained.We study the propagation and collision properties between two solitons,and construct the rogue-wave solutions.We analyze the characteristics of the solitons and rogue waves.In chapter 3.we study a variable coefficient Ablowitz-Ladik equation.Using the Hirota method,the dark scoliton solutions are studied.Employing the similarity trans-formation and KP hierarchy reduction,we obtain the rogue-wave solutions in terms of the Gramian under certain variable-coefficient,constraintsWe graphically study the rogue waves with the effects of the coefficients.In chapter 4,we study a(2+1)-dimensional Ablowitz-Ladik equation.We obtain the bilinear forms and the bright and dark soliton solutions.We also discuss the in-teractions of the solitons.Employing the KP hierarchy recduction,we obtain the rogue wave solutions in terms of the Gramian.We graphically study the first-,second-and third-order rogue waves.In chapter 5,we study a(3+1)-dimensional generalized B-type KP equation.Em-ploying the Hirota method and symbolic computation.we obtain the lump,breather-wave and rogue-wave solutions under certain constraints.We graphically study the lump and rogue waves.In chapter 6,we study a variable-coefficient KP equation.Employing the KP hier-archy reduction,we obtain the rogue-wave solutions in terms of the Gramian.Periodic and s-shaped line rogue waves are presented with the different values of the dispersion coefficient.The second-order rogue waves and multi-rogue waves are also graphically discussed.In chapter 7,we study an inhomogeneous discrete nonlinear Schrodinger equation.Via the binary Bell polynomials,we construct the bilinear forms,bilinear Backlund transformations and discrete,soliton solutions for the equation.Employing the KP hierarchy reduction,we obtain the rogue-wave solutions.We graphically study the solitons and rogue waves with the effects of the coefficients.In chapter 8,we summarize the main contents and innovations of this dissertation,and present our further research works.