Research On The Symmetries And Analytical Solutions To Some Nonlinear Differential Equations

In this thesis,we study the symmetry,the analytical solutions for several types of nonlinear differential equations.Firstly,the related background and the main work of this thesis are briefly intro-duced in the first chapter.And then,based on the Lie symmetry method,we investigate the generalized Boussinesq equation in the second chapters.Its infinitesimal symmetric vector fields,the adjoint representations and group invariant solutions are obtained,the solutions transform the original equation into ordinary differential equation.Based on this,the exact analytical solutions and the exact power series solutions of the equation are obtained by power series theory and comparison coefficient method.In the third chapter and fourth chapter,based on the Hirota bilinear method,the Bell polynomials and Riemann theta function are extended to the(2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation,result in its bilinear form and analytic solutions,which includes soliton solutions and periodic wave solutions.Fur-thermore,the asymptotic behavior of the periodic wave solutions is studied.We prove that the periodic wave solutions and soliton solutions are closed related,that is,the pe-riodic wave solutions can degenerate into soliton solutions under some limit condition,and finally we give a graphical simulation and analysis by Maple software.In the fifth chapter,in the same way,based on the Hirota bilinear method and Riemann theta function,we investigate the quasi-periodic waves solutions?soliton solutions and asymptotic properties for a generalized(3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation,the propagation diagram of pe-riodic wave solution is drawn by Maple software.whatever,we also using hyperbolic function the tanh method and tan method to establish more traveling wave solutions of the VC-BKP equation.Finally,we give some summarizes and prospects of this thesis.