Quasi-periodic solutions of the Heisenberg hierarchy

TL;DR

This paper derives the Heisenberg hierarchy's Hamiltonian structure, introduces algebraic curves and meromorphic functions, and provides explicit theta function solutions, advancing the understanding of integrable systems in mathematical physics.

Contribution

It presents a novel derivation of the Heisenberg hierarchy's structure and explicit solutions using algebraic geometry and theta functions.

Findings

Explicit theta function representations of solutions

Algebraic curve of genus n associated with the hierarchy

Hamiltonian structure derived via zero curvature and trace identity

Abstract

The Heisenberg hierarchy and its Hamiltonian structure are derived respectively by virtue of the zero curvature equation and the trace identity. With the help of the Lax matrix we introduce an algebraic curve $\mathcal{K}_{n}$ of arithmetic genus $n$, from which we define meromorphic function $\phi$ and straighten out all of the flows associated with the Heisenberg hierarchy under the Abel-Jacobi coordinates. Finally, we achieve the explicit theta function representations of solutions for the whole Heisenberg hierarchy as a result of the asymptotic properties of $\phi$.