Study On The Generation And Properties Of Several Integrable Systems

This dissertation investigates generation and related properties of several kinds of integrable systems in non-linear science.The main work is as follows: Loop algebra and Tu-Scheme methods are used to generate isospectral and non-isospectral integrable equations hierarchy and a conservation law of one of the equations;(1 +1)-dimensional,(2 + 1)-dimensional discrete integrable hierarchy and their extended integrable hierarchy are generated using the Tu-scheme;the R-matrix method is applied to construct Toda lattice systems and Extended discrete systems;a class of generalized shallow water wave equations are obtained by means of symmetric reduction,and their Lax pairs,symmetric,invariant solutions,and sequential solutions and their corresponding self-adjoint systems and conservation laws are obtained;similarity solutions and numerical solutions of time-fractional Burgers system are given.Numerical simulation and error estimation are given.In Chapter 1,an introduction is devoted to review the research background and current situation of nonlinear science and integrable systems,the important integrable system generation methods in mathematical physics,several methods for seeking exact solutions of the integrable system,the research background and current situation of fractional partial differential equations.Finally,the main work of this dissertation is clarified.In Chapter 2,the Tu-scheme method is used to generate several kinds of continuous and discretitye integrable systems.In the first section,we obtain isospectral and non-isospectral Lax pairs,and generate isospectral and non-isospectral integrable hierarchy by the Tu-scheme.In the second part of this chapter,we still use the Tu-scheme to obtain the(1 + 1)-dimensional and the(2 + 1)-dimensional discrete integrable systems.We also use the potential function to obtain a new difference-differential equation.In the third part,we find the Hamilton structure,genetic operator and symmetry of these integrable systems.In addition,we have established the B¨aclund transform of the isospectral equation hierarchy.A kind of isospectral equation is reduced to a new longwater wave equation.The Lie Group method is used to find similarity solutions,nonsimilarity solutions and non-linear self-adjoint solutions of these equations.Finally,we use the Variational method to analyze the infinite conservation law of the long-water wave equations.In Chapter 3,the R-matrix method is applied to derive a Toda lattice system that is widely used in statistical physics and quantum physics.First,a new discrete integrable system generation formula is constructed using the R-matrix,and the extended Toda lattice system and its Lax pair are obtained.Then we use this formula again to get the corresponding(2 + 1)-dimensional Toda lattice system and its extended discrete system,and find their Lax pair.Finally,we get the infinite conservation laws of a(1+1)-dimensional generalized Toda lattice system and a new(2 + 1)-dimensional lattice system.In Chapter 4,we reduce a class of generalized long water wave system to a standard water wave system,and further obtain Lax pairs,symmetric,invariant solutions,and sequence solutions of generalized shallow water waves.In addition,we also study the corresponding self-adjoining systems and conservation laws of the long-water wave system.In Chapter 5,the similarity solutions of the time fractional Burgers system are discussed.Using Lie-point symmetry and variational method,the fractional partial differential equations are transformed into Riemann-Liouville type fractional ordinary differential equations,and the similarity solutions and numerical solutions equations are obtained.In addition,the scale transformation is used to transform the fractional partial differential equation into a fractional ordinary differential equation in the Caputo sense.We find that its solution can be expressed by the ? function.Finally,we also get this approximation Numerical solution of the equation.In Chapter 6,the primary of this dissertation and further work is prospected.There are 3 figures,3 tables and 169 references in this dissertation.