Nonlocal Symmetries And Explicit Solutions Of The Nonlinear Evolution Equation

In this paper, residual symmetry, nonlocal symmetry and Lie point symmetry are ap-plied to reduce some nonlinear evolution equations. And the main results as following:Firstly, the residual symmetry of the (2+1)-dimensional modified dispersive water-wave system and C-K direct method are employed to reduce the equation to ordinary differential equation. And based on it, some new soliton solutions of exact significance are obtained.Secondly, the nonlocal symmetry which contains Lie point symmetry of the coupled nonlinear Schrodinger equation and Lie point symmetry reduction are applied to reduce the equation to ordinary differential equation. Then, by solving the ordinary differential equation, a new solution of the equation is achieved.The dissertation is structured as follows:Section 1, briefly describe the paper’s background, research significance and introduce the description and research status of three methods and some basic knowledge about this paper.Section 2, the standard Painleve truncate expand is used to find the residual symmetry of (2+1)-dimensional modified dispersive water-wave system. And based on the above sym-metry, we discuss some transformation invariance. Then, the C-K direct method is used to reduce symmetry and obtain several kinds of new solutions.Section 3, based on the coupled nonlinear Schrodinger equation and its known Lax pair, we obtain the nonlocal symmetry of the equation. By defining a new variable, localized the nonlocal symmetry to a Lie point symmetry which has a closed prolonged system. And based on it, the explicit analytic interaction solution related to Jacobi elliptic function is derived. Figures show some physical interaction between cnoidal waves and a solitary wave.Section 4, summarized the work of this paper and to make a further research direction in future.