Decomposition Of Reduced Semi-Discrete Chen-Lee-Liu Equations And Quasi-Periodic Solutions

Starting with a2x2matrix eigenvalue problem, a hierarchy of nonlinear differential-difference equations which contains reduced semi-discrete Chen-Lee-Liu equation as its first nontrivial equation is proposed. Then, a new symplectic map and a class of finite-dimensional Hamiltonian systems are obtained with the aid of the nonlinearization ap-proach. Further, according to the generating function approach, the involutivity and the functional independence of the conserved integrals are proved. Hence, the Hamiltonian sys-tems being completely integrable in the Liouville sense are verified. Based on the theory of algebraic curves, Abel-Jacobi coordinates are introduced to straighten out the continuous flow and discrete flow. Quasi-periodic solutions are obtained with the help of the Riemann theta functions.