Finite Genus Solutions Of Several Purely Discrete Integrable Equations

This thesis is dedicated to three AKNS discrete integrable equations, the Hirota (lattice sine-Gordon) equation, the lattice potential MKdV equation (H30 model), and the Q10 model, where H30, Q10 model are the special H3, Q1 model, respectively, in the ABS (2003) list with the parameterδvalues zero; finite genus solutions of them are calculated through "decomposition, straightening out, in-version" three steps, respectively. Take an appropriate Darboux transformation of the continuous spectral problem as the discrete spectral problem, construct the Lax representations of the discrete integrablc equations. Under certain constraints between the potentials and the cigenfunctions, the continuous spectral problem is nonlinearized into an integrable Hamiltonian system, while the discrete spectral problems into integrable symplectic maps; they share the same Liouville set of integrals and. the corresponding flows commute. By introducing the appropriate Abel-Jacobi variables, these flows are straightened out in the Jacobi variety of the spectral curve, which is defined by the Lax matrix, with the constant angu-lar velocities. In the Abel-Jacobi coordinates, a special solution is represented as a linear combination of these vectors; through Riemann-Jacobi inversion, finite genus solution expressed by theta functions in the original potential variables is derived. The Hirota equation and the LpMKdV equation can be mapped into each other by Nijhoff's transformation, their solutions can be gained via maps Sβ, Sγor Sβ1, Sβ2, two ways; any Mobius transformation with constant coefficients of a solution of the Q10 model is also a solution.