Trigonal curves and algebro-geometric solutions to soliton hierarchies

TL;DR

This paper develops a framework connecting trigonal algebraic curves with soliton hierarchies, providing explicit algebro-geometric solutions using Riemann theta functions and analyzing the AKNS hierarchy.

Contribution

It introduces a novel approach to construct algebro-geometric solutions for soliton hierarchies via trigonal curves and extends the theory to arbitrary hierarchies.

Findings

Derived Dubrovin equations for zeros and poles.

Straightened flows in soliton hierarchies using Abel-Jacobi coordinates.

Constructed explicit solutions with Riemann theta functions.

Abstract

Using linear combinations of Lax matrices of soliton hierarchies, we introduce trigonal curves by their characteristic equations, and determine Dubrovin type equations for zeros and poles of meromorphic functions defined as ratios of the Baker-Akhiezer functions. We straighten out all flows in soliton hierarchies under the Abel-Jacobi coordinates associated with Lax pairs, and generate algebro-geometric solutions to soliton hierarchies in terms of the Riemann theta functions, through observing asymptotic behaviors of the Baker-Akhiezer functions. We analyze the four-component AKNS soliton hierarchy in such a way that it leads to a general theory of trigonal curves applicable to construction of algebro-geometric solutions of an arbitrary soliton hierarchy.