Binary Bargmann symmetry constraint and algebro-geometric solutions of a semidiscrete integrable hierarchy

Abstract

We present the binary Bargmann symmetry constraint and algebro-geometric solutions for a semidiscrete integrable hierarchy with a bi-Hamiltonian structure. First, we derive a hierarchy associated with a discrete spectral problem by applying the zero-curvature equation and study its bi-Hamiltonian structure. Then, resorting to the binary Bargmann symmetry constraint for the potentials and eigenfunctions, we decompose the hierarchy into an integrable symplectic map and finite-dimensional integrable Hamiltonian systems. Moreover, with the help of the characteristic polynomial of the Lax matrix, we propose a trigonal curve encompassing two infinite points. On this trigonal curve, we introduce a stationary Baker–Akhiezer function and a meromorphic function, and analyze their asymptotic properties and divisors. Based on these preparations, we obtain algebro-geometric solutions for the hierarchy in terms of the Riemann theta function.