A New Integrable Symplectic Map And The Exact Solution Associated With Nonlinear Lattice Equations

The thesis mainly studies the integrability and the binary nonlinearization under Bagmann symmetry of discrete integrable systems. A family of finite-dimensional completely integrable systems and a new integrable symplectic map are provided in terms of the binary nonlinearity of spectral problem. After that, the solution of the discrete lattice equation can be gained by the way of the Lie point symmetry theory.In the first chapter, the origin and development of soliton theory and its research status and the background of study and application are summarized. The integrable systems and the integrable couplings are introduced in detail.In the second chapter, the basic knowledge needed in the study of this subject is introduced. We first understand the definition of two meanings of integrablility——Liouville integrability and Lax integrability. The trace indentity of discrete integrable systems and the Tu scheme of discrete spectral problems are given. Then the binary nonlinearization under Bagmann symmetry of discrete integrable systems is also introduced in detail. Finally, we described in detail the two concepts——classical Lie group method and The modified CK direct method.In the third chapter, A new discrete integrable hierarchy and its Hamiltonian structure are studied. The third chapter divided into two parts: In the first part, by choosing a discrete spectral problem, a family of differential-difference equation is derived from the discrete zero curvature equation. And its Hamiltonian system is established. In the second part, the Liouville equation is produced by the trace indentity of discrete integrable systems.In the fourth chapter, a new integrable symplectic map is studied,and we use the symmetry theory to find the solution. The fourth chapter divided into two parts: In the first part, then the Lax pairs and the adjoint Lax pairs are nonlineared under a proper symmetry constraint. Their temporal part and the spatial part are severally nonlinearized into a new integrable symplectic map and the a finite-dimensional integrable system.In the second part, based on one parameter group of transformations, the continuation vector field could be obtained by using the infinitesimal generator. Then the solution could be solved according to the new infinitesimal generator, which could be obtained by substituted the new continuation vector field into original equations.