Integrability And Solutions Of Some Nonlinear Differential Equations

With the rapid development of nonlinear science. the number of nonlinear differential equa-tions is also increasing. In the description of physical phenomena. nonlinear differential equation is an important mathematical model, and one of the most popular research topics. The research on the integrability and exact solution of nonlinear differential equation, in theory, helps to understand the movement rules of various substances under the nonlinear interaction in the physical phenomena, and thus helps to advance its practical application.The research on the exact solution is the primary task to discuss the problem of nonlinear differential equation. Many research topics. such as the Painleve test, constructing multi-soliton solutions, and searching for wave solutions etc., often involve a large amount of tedious algebra reasoning and symbolic computation, which can be difficult to implement in practice. Thus the research progress of nonlinear differential equations is affected. In recent years, due to the development of symbolic computation, the study of nonlinear differential equation has been greatly promoted, and the results of their research, especially new solutions emerge in large numbers.By means of symbolic computation system Maple, this dissertation mainly studies three types of nonlinear differential equations, which are extended Euler top. B-type KdV equations and nonlinear Schrodinger equation with variable coefficients. The integrability is discussed and exact solutions are obtained, the main work is as follows:Chapter1is concerned with the backgrounds and development of nonlinear differential equation, soliton theory and integrable theory, then takes simple overview of several methods for solving nonlinear differential equation and types of integrability. In addition. a brief description of main work and structural arrangements in this thesis is shown.Chapter2first introduces the classic Euler top. which as a starting point. the extended Euler top is constructed by using the generalized Lax pair. Then the Painleve integrability of extended Euler top is judged by Painleve test. and the exact solutions are calculated under Liouville theorem and conserved quantities.Chapter3mainly studies B-type KdV equations. first verifies the Painleve integrability. then derives Backlund transformation by means of Painleve truncation method. Further more the bilinear formulation. bilinear Backlund transformation and soliton solutions of the equations are constructed with the aid of Hirota bilinear method.In Chapter4. a class of nonlinear Schrodinger equation with variable coefficient is re-searched. of which Painleve integrability is judged by symbolic computation. Then Darboux transformation is constructed on the basis of Lax pair. Filially a new solution is constructed from the zero solution under Darboux transformation.At last, a short summary of the dissertation is given.