Exact Solutions Of Nonlinear Evolution Equation (s) With The Painleve Analysis

In this paper, some important nonlinear evolution equations have been studied. A summary of the main conclusions of our researches is in the following:(1) Nonlinear partial differential equation (NPDE) is converted into an ordinary differential equation (ODE) via a new anstaz. Using undetermined function method, the exact solutions and solitary solutions of the NPDE are obtained by solving the ODE. Burgers equation and Whitham-Broer-Kaup shallow water equations are chosen to illustrate this method.(2)By the Jacobi elliptic function expansion method and computersymbolic systems (Maple, Matlab or Mathematica), many exact periodic solutions of nonlinear wave equations can be obtained and under some conditions, these solutions degenerate solitary wave solutions, shock wave solutions, trigonometric function solutions respectively. We improve this method as follows: (1)Single Jacobi elliptic function is replaced with an unified Jacobi elliptic equation, thus repeated calculation is avoided; (2)Extending the expansion method from sole-function to double- function form, the more solutions for NPDE are obtainted; (3) Using many of Jacobi elliptic functions besides ordinary three kinds, the content of solutions represented by Jacobi ellipticBy the Jacobi elliptic function expansion method and computerfunctions is very abundant. BBM equation and Boussinesq equations are chosen to explain our method.(3) For the coupled KdV equations ut+vvx+uux+uxxx=0,vt+(uv)x+vvx=0,it is shown that this equations possess Painleve property, thus it is P-integrable. Furthermore an auto-Backlund transformation is given. By auto-Backlund transformation, the solitary solutions of the coupled KdV equations are obtained from its trial solution.(4) By the application of the truncated W. T. C mehtod, soliton-like solutions for a (2+1) dimensional generalized Boussinesq equation utt-uxx-uyy-(u2)xx-uxxx=0 found.