Symmetries For Some Discrete Integrable Systems And Exact Solutions To The Inverse AKNS Equation

The major contents in this dissertation include:symmetries and Lie algebra for the discrete isospectral equation hierarchy related to a general pseudo-difference operator; Analysis of Casoratian solutions to the isospectral differential-difference Kadomtsev-Petviashvilli(KP) equation and its connection with symmetries;Soliton solutions for negative isospectral AKNS equation and negative nonisospectral AKNS equation are studied by using Hirota method;new integrable decompositions of AKNS equation,Gauge transformation, binary constraint,integrable decompositions,infinite conservation laws of Geng equation.In the second chapter,Some general results of integrability and algebraic structures for hierarchies related to a general pseudo-difference operator are presented.These general results include general expression formulas of isospectral hierarchy and non-isospectral hierarchy, explicit Lax pairs of these hierarchies,Lie algebraic structure of these hierarchies, two set of symmetries of the isospectral equations and their Lie algebraic structure.Taking the advantage of reduction relationship,the discrete Gel'fund-Dickey Hierarchy and its K-symmetries are presented.The Casoratian solutions and the corresponding Casoratian conditions for the isospectral differential-difference Kadomtsev-Petviashvilli(KP) equation are disscussed,a more general Casoratian condition is obtained and it is proved that this general Casoratian condition can not derive new solutions.It is interesting that the relationship between Casoratian conditions for the isospectral differential-difference KP equation and the invariance of this equation is found,which show one symmetries of the isospectral differential-difference KP equation,a 3-dimensional closed Lie algebra are further obtained.In the fourth chapter,the solition solutions of negative isospectral AKNS equation and nonisospectral AKNS equation are presented.The dynamical character of the 2-soliton solution to the negative isospectral AKNS equation are discussed,it is found that the interactions of the 2-solition shows the usual phase differences and the changing amplitudes.The fifth chapter is mainly focused on studying a new approach to obtain new integrable decompositions.A new approach to construct a new m×m matrix spectral problem from a normal 2×2 matrix spectral problem is presented.AKNS spectral problem is discuss as an example,the isospectral evolution equations of the new m×m matrix spectral problem is nothing but the famous AKNS equation hierarchy.With the aid of the binary nonlinearization method,we get the new integrable decompositions of the AKNS equation.Three new integrable decompositions and the conservation laws of the Geng equation are also obtained.The binary constraints of the Geng hierarchy obtained with the use of Gauge transformation is first proposed.