Symmetry,Exact Solutions For Several Kinds Of Nonlinear Partial Differential Equations And Integrable Systems

This dissertation investigates symmetry,exact solutions of several kinds of nonlinear partial differential equations in nonlinear mathematical physics and integrable systems.The main work is carried out in four aspects: the group classification method is applied to study a modified sixth-order thin film equation,the Lie group method is developed to investigated fractional partial differential equations,Bell polynomials are used to construct bilinear forms and soliton solutions and Riemann theta functions are used to construct quasi-periodic wave solutions for some nonlinear partial differential equations directly,the generating of integrable system and its Hamilton structure.In Chapter 1,an introduction is devoted to review the research background and the current situation related to this dissertation,including symmetry theory,several classical methods for seeking exact solutions of nonlinear partial differential equations and the theory of integrable systems.The main work of this dissertation is also illustrated.In Chapter 2,a group classification is preformed on a modified sixth-order thin film type equation under the group of continuous equivalence transformation.In order to get all the inequivalent similarity reductions,the one-dimensional optimal systems of subalgebras are constructed.A list of similar reductions is presented based on the optimal systems.Some invariant solutions with physical interest are obtained.The nonlinear self-adjointness of the equation is showed and conservation laws are obtained by using the new conservation theorem proposed by Ibragimov.In Chapter 3,the Lie group method is developed to investigate the time fractional foam drainage equation,the time fractional Derrida-Lebowitz-Speer-Spohn equation and the time fractional thin film equation with Riemann-Liouville derivative.Firstly,we derive Lie symmetries of the time fractional foam drainage equation and transform it into a nonlinear ordinary fractional differential equation with Erdélyi-Kober derivative.Conservation laws of the equations are also constructed by using the symmetries we obtained.Secondly,we investigate Lie symmetries admitted by the time fractional Derrida-Lebowitz-Speer-Spohn equation.In a particular case of scaling transformations,we transform the equation into a nonlinear ordinary fractional differential equation.We further consider the nonlinear self-adjoint condition of the equation and construct conservation laws with the aid of the new conservation theorem and the fractional generalization of the Noether operators.Finally,we derive classical symmetries admitted by the time fractional thin film equation involving an arbitrary function f.The one-dimensional optimal systems are determined and similar reductions are performed accordingly.All inequivalent reduced fractional ordinary differential equations are constructed and some invariant solutions are obtained.In Chapter 4,based on the Hirota bilinear method,we construct soliton solutions and quasi-periodic wave solutions for some nonlinear partial equations.Firstly,we construct new bilinear forms and bilinear B?cklund transformations for the associated Camassa-Holm equation and its integrable generalization by using binary Bell polynomials.Soliton solutions of the equations are obtained and interaction behaviors of the solitons are illustrated graphically.Secondly,the binary Bell polynomials method is applied to the generalized variable-coefficient fifth-order Kd V equation.We derive bilinear form of the equation,from which we can get soliton solutions directly.With the aid of multidimensional Riemann theta functions,the Hirota bilinear method are extended to construct quasi-periodic wave solutions of the equation.The asymptotic properties of quasi-periodic wave solutions are analyzed and the relation between the quasi-periodic wave solutions and the soliton solutions is established.Finally,we investigate quasi-periodic wave solutions for the Kersten-Krasil'shchik coupled Kd V-m Kd V system,which is more difficult to be dealt with than single equation due to the appearance of two equations and two functions.Through proper variable transformations,the system under consideration is transformed into an appropriate bilinear form for constructing quasi-periodic wave solutions.Based on the bilinear form and multidimensional Riemann theta functions,we construct quasi-periodic wave solutions of the system.The asymptotic properties of the quasi-periodic wave solutions are also proved.In Chapter 5,the method of generating integrable systems and their Hamilton structure is studied.We present a generalized Li spectral problem based on the three dimensional real special orthogonal Lie algebra so(3,R),and derive a generalized Li hierarchy by employing the Tu scheme.The Hamilton structure and Liouville integrability of the system are also obtained with the aid of the trace identity.In Chapter 6,the primary of this dissertation are summarized and further work is prospected.