Integrability And Spectral Stability Of Solitary Waves For Several Kinds Of Nonlinear Wave Equations

Nonlinear waves exist in nature,such as water waves,shock waves in the gas,plas-ma waves,shock waves in solids,density waves in galaxies,seismic waves and so on.The nonlinear wave equations are important nonlinear mathematical physics models for describing fluctuation phenomena.Through the research about the integrability,finding solutions and the dynamic behavior of solitary waves of nonlinear wave equations can help us reveal the propagation law of nonlinear waves,explain the corresponding natural phenomena,and also promote the development of nonlinear wave theory.This dissertation focuses on the integrability,solitary wave solutions and their dynamics for the BKP equation,(2+1)-dimensional KdV equation,KP-Based system,Sharma-Tasso-Olver equation.These equations have a wide range of applications in flu-id mechanics,plasma physics and nonlinear optics.The main themes of the dissertation include:1.Based on the binary Bell polynomial theory and Hirota bilinear method,the bi-linear form and solitary wave solutions of BKP equation are studied.By introducing the proper differential constraint condition and decoupling technique,several kinds of Hiro-ta bilinear differential equations,Bell-polynomial-type Baklund transformations and Lax pairs are obtained for the BKP equation.With the aid of the(3+1)-dimensional Hirota bilinear equation,the multi-wave solution,complexiton solution,bright-dark lump solu-tions and kink-lump wave interaction solutions are presented.The relationship between complexiton solution and bright-dark lump wave solution is investigated.It is found that the bright-dark lump is the limitation of complexiton solution,and the analytic expres-sion of complexiton-solution can be derived from the power series of the trigonometric function csc2(?x).In addition,the infinite conservation laws of the BKP equation are constructed by the Bell-polynomial-type Lax pairs.2.The symmetry,KMV type Lie algebra and conservation laws of BKP equation are presented by using the generalized symmetry method.Based on the obtained symmetry structure form,the generalized group-invariant solution for the BKP equation is directly constructed.By using the generalized group-invariant solution,we derive the continu-ous symmetry group and discrete symmetry group for the BKP equation.The non-auto Baclund transformation and nonlocal symmetry for the BKP equation are obtained by the truncated Painleve method.3.The lump waves for(2+1)-dimensional KdV equation and KP-Based system are investigated by the Hirota bilinear method.The numerical simulation research shows that these lump waves possess the spatio-temporal deformation characteristics.In the differ-ent parameter regions,the lump wave solution will present the different spatio-temporal structures.Theoretical analysis indicates that the bifurcation of the equilibrium point is one of the reason that causes this phenomenon.4.Based on the Hirota bilinear method and planar dynamical systems method,the existence of the kink wave solution for STO equation is given.The spectral stability of kink wave solution is shown by the spectral energy estimate method.By using the extended homoclinic test function method,a class of kink wave interaction solution is presented.The numerical simulation and theoretical analysis show that the fusion and fission phenomena of this kind of kink wave solution of the STO equation don't depend on the dispersion coefficient a,they are determined by the translation parameter(?).The sign of a determines the direction of propagation of the solitary wave:When ?<0,the solitary wave propagates to the left;when a>0,the solitary wave propagates to the right.