Quasi-periodic Solutions Of The Soliton Hierarchies Associated With2×2Discrete Matrix Spectral Problems

In the past few years a gradual shift of emphasis has taken place from con-tinuous to discrete integrable systems, many nonlinear integrable lattice equations have been proposed and discussed, for example, the Ablowitz-Ladik lattice, the Toda lattice, and so on. Such systems arise and play an important role in a very large number of contexts and have an extensive range of applications in mathemat-ical physics, statistical physics, disordered systems, biology, economics, numerical analysis, discrete geometry, cellular automata, quantum field theory and so on. It is well known that quasi-periodic (or finite-band, or algebro-geometric) solutions of soliton equations reveal inherent structure mechanism of solutions and describe the quasi-periodic behavior of nonlinear phenomenon or characteristic for the integra-bility of soliton equations. In the theory of integrable systems, algebro-geometric method offers an approach to seek quasi-periodic solutions, which can be explicitly given by θ functions on the Riemann surface.This paper mainly discussed about the quasi-periodic solutions of several hier-archies of soliton equations with clear physical meaning which associate with2×2matrix spectral problems by means of the algebro-geometric method. These hier-archies associated with the2×2matrix spectral problems we consider here are the discrete mKdV hierarchy, the Relativistic Toda hierarchy, the discrete self-dual network hierarchy and the new differential-difference hierarchy.First, we introduce the Lenard recursion gradients and derive the hierarchy that associated with the2×2matrix spectral with the aid of the zero-curvature equation. Then, we introduce a Lax matrix and establish a direct relation between a direct relation between the elliptic variables and the potentials. The correspond-ing hierarchy is separated into solvable ordinary differential equations. Next, the hyperelliptic Riemann surface Kn of arithmetic genus N and the Abel-Jacobi co-ordinates are brought in view of the characteristic polynomial of Lax matrix, from which the associated meromorphic function are given and the corresponding various flows are straightened out, including the continuous and discrete ones. finally, we define the Baker-Akhiezer function with the aid of the associated meromorphic func-tion and construct three kinds of Abel differentials. By analysing the asymptotic properties of the three kinds of Abel differentials, the meromorphic functions and Baker-Akhiezer functions, we obtain the explicit Riemann θ function representations of the meromorphic function, the Baker-Akhiezer function, and in particular, that of the potentials for the entire hierarchy.