Integrable Systems And Solutions To Noniospectral Soliton Equations

The major contents in this dissertation include: the generation of soliton hierarchies of equations and the structure equations of Lie group, Hamiltonian structures, Liouville integrability, infinite conservation laws, binary nonlinearization of Lax pairs and adjoint Lax pairs and integrable symplectic map and finite-dimensional integrable systems, expanded integrable models of soliton equations, multi-soliton solutions of some isospectral and nonisospectral soliton equations are studied by using Hirota method, Wronskian technique, soliton equations in 1+1 dimensions are generated by using symmetry constraints of soliton systems in 2+1 dimensions, completely conditions for symmetry constraints of potentials are constructed by means of the relation of the Gateaux derivative and funca-tional derivative.In the second chapter, a new isospectral problem is presented and a hierarchy of Lax integrable soliton equations are derived from the spectral problem. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structures. The corresponding Lax pairs and adjoint Lax pairs are nonlinearized into finite-dimensional integrable Hamiltonian systems and a new integrable symplectic map. Involutive representations of solutions of soliton equations is given by involutive solutions of commutative flows. Finally, expanding integrable models of the hierarchy are constructed by using a new Loop algebra G.Three discrete isospectral problems are investigated in the third chapter. Firstly, a family of lattice soliton equations are derived from a discrete isospectral problem. It is verified that the hierarchy possesses discrete Hamiltonian structure and is integrable in the Liouville sense. New finite-dimensional integrable Hamiltonian systems and an integrable symplectic map are generated by binary nonlinearization method. Conservation laws are established. secondly, a hierarchy of discrete equations associated with Lotka-Volterra are derived from a new algabraic system. Its integrability and integrable couplings are studied. Lastly, positive and negative hierarchies of discrete soliton equations are derived from a spectral problem. An explicit symmetry constraint is proposed. Lax pairs are nonliearized into a new integrable symplectic map and finite-dimensional Hamiltonian systems.In the fourth chapter, nonisospectral AKNS equations are derived from the structure equations of Lie group. By taking a Loop algebra, expanded integrable models of nonisospectral AKNS equations are established. The bilinear derivative equations and the N-...