Study On The Related Problems Of Soliton,Integrable Systems,and Fractional Differential Equations

Soliton equations are mathematical models arising in physical problems or phenomenon.In the field of mathematics,it promotes the development of mathematics,such as the application of Lie group to the differential equation,also makes people understand soliton more profoundly,and more models were put forward to study the related properties.Soliton theory,as an important component of the applied mathematics and mathematical physics,its research contents and methods are very rich,such as proposing soliton equations with geometry,using the theory of Lie algebra to study integrable systems and using numerical analysis methods to research solutions of soliton equations.In recent years,domestic and foreign scholars pushed forward the theory from different angles in different styles.This thesis mainly investigates integrable systems with the reduction of the self-dual Yang-Mills equations,generates integrable equations by symmetric space and homogeneous space and the representation of Riemann curvature tensors,and constructs integrable lattice hierarchies by some matrices and operator Lie algebras,quasi-Hamiltonian form and Darboux transformation.This thesis also studies nonlinear evolution equations through symmetry and other methods.Finally,fractional derivatives and local fractional derivatives are used to study fractional differential equations and their applications are elcudidated.The thesis consists of five chapters.Chapter 1 introduces the historical background,research significance and the present development of soliton.Some basic knowledge and the main work done in this thesis are also summarized.In chapter 2,the problem of the integrable system is studied.In section 2,through imposing space-time symmetries,a new reduction of the self-dual Yang-Mills equations is obtained for which a Lax pair is established.By using a proper exponential transformation,we transform the Lax pair to get a new Lax pair whose compatibility condition gives rise to a set of partial differential equations.In section 3,we first introduced a linear stationary equation with a quadratic operator in?_x and?_y,then a linear evolution equation is given by N-order polynomials of eigenfunctions.A second-order flow associative with a homogeneous space is derived from the integrability condition of the two linear equations.Finally,as an application of a Hermitian symmetric space,we establish a pair of spectral problems to obtain a new(2+1)-dimensional generalized Schr?dinger equation.In section 4,two types of matrix Lie algebras are presented and two kinds of loop algebra are established.We make use of the first loop algebra to obtain a new(1+1)-dimensional integrable discrete hierarchy.Next,we apply the second matrix loop algebra to introduce an isospectral problem and deduce a new integrable discrete hierarchy,whose quasi-Hamiltonian structure is derived from the trace identity proposed by Tu Guizhang.A type of Darboux transformation of a reduced discrete system of the latter integrable discrete hierarchy is obtained as well.We introduce two types of operator Lie algebras according to a given spectral problem by a matrix Lie algebra and apply the r-matrix theory to obtain a few new lattice integrable systems,including two(2+1)-dimensional lattice systems.Chapter 3 deals with the problem of symmetric reduction of the nonlinear evolution equations.In section 1,the classical Lie group method and the generalized symmetry method are adopted to analyze a(2+1)-dimensional generalized PainlevéBurgers system.In section 2,the continuous classical Lie group method is extended to differential-difference equations.Lie group technique,the symmetry reduction method and the rational expansion method are adopted to study differential-difference equations.The reduced relativistic Toda lattice system is used as an example to elucidate the solution process.Chapter 4 studies the solutions of the fractional differential equation and its applications.The definition of the fractional derivative,the change of fractal space to its continuous partner,basic laws of mechanics in fractal space and methods of solving fractional differential equations are introduced.Through the fractional complex transformation,the modified exponential function is used to solve the fractional differential equations,such as the fractional Benjamin-Bona-Mahony(BBM)equation,Whitham-Broer-Kaup(WBK)equation and Hirota-Satsuma(HS)equation.The two modified KdV equations are solved by the variational iteration method within local fractional derivative.Finally,the main conclusions of the thesis and research prospects are given in chapter 5.