On Analytical Solutions,Integrability And Riemann-Hilbert Problem To Nonlinear Differential Equations

In this thesis,we investigate the analytical solutions,symmetry,conservation laws and Riemann-Hilbert problem for several types of nonlinear differential equations.In Chapter 2 and 3,based on the Hirota bilinear method,the Bell polynomials and Riemann theta functions are used to investigate the(3+1)-dimensional KdV-like mod-el.Its bilinear representation,Backlund transformation,Lax pair,infinite conservation laws are presented with a detailed analysis.Based on its bilinear form,the soliton solu-tions and periodic wave solutions of the equation are also constructed by using Riemann theta function.Moreover,the relationship between periodic wave solutions and soliton solutions are systematically established.In chapter 4,Lie symmetry analysis method is used to explicitly study the gen-eralised Whitham-Broer-Kaup-Like equation.Its vector fields,optimal system,sym-metry reductions and group invariant solutions are obtained.Then the power series solutions are also derived by using the power series theory.Moreover,based on a new theorem of conservation laws,the conservation laws associated with symmetries of this equation are constructed with a detailed derivation.In chapter 5,the(2+1)-dimensional Ito equation and a generalized(3+1)-dimensional Kadomtsev-Petviashvili equation equation are systematically investigated.Based on their bilinear forms,the breather wave solutions and rogue wave solutions of the equations are well constructed.Moreover,the dynamic behaviors of breather waves and rogue waves are analyzed with some graphics,respectively.Then,based on the bilinear formalism of a new generalized(3+1)-dimensional Kadomtsev-Petviashvili e-quation,a direct method is employed to explicitly construct its mixed solutions with an ansatz function.Finally,the interaction phenomena between rogue waves and soli-tary waves are analyzed with some graphics.In addition,following a similar method with the(2+1)-dimensional Ito equation,a generalized(2+1)-dimensional Boussinesq equation are investigated.In what follows,Riemann-Hilbert approach is used to succinctly investigate Riemann-Hilbert problem and multi-soliton solutions of the mixed coupled nonlinear Schrodinger equations.Furthermore,the dynamics of solitary waves are analyzed with some graphics.Finally,we give some summarizes and prospects of this paper.