Dubrovin equations and integrable systems on hyperelliptic curves

TL;DR

This paper generalizes Dubrovin equations for divisors on hyperelliptic curves and demonstrates their role in linearizing integrable flows, revealing a fundamental principle linking these equations to hierarchies of soliton equations.

Contribution

It introduces the most general Dubrovin-type equations for hyperelliptic curves and establishes their equivalence with integrable soliton hierarchies, providing a new framework for understanding these systems.

Findings

Unified framework for Dubrovin equations and soliton hierarchies

Linearization of integrable flows on hyperelliptic curves

Encoding of soliton hierarchies via Dubrovin equations and trace formulas

Abstract

We introduce the most general version of Dubrovin-type equations for divisors on a hyperelliptic curve of arbitrary genus, and provide a new argument for linearizing the corresponding completely integrable flows. Detailed applications to completely integrable systems, including the KdV, AKNS, Toda, and the combined sine-Gordon and mKdV hierarchies, are made. These investigations uncover a new principle for 1+1-dimensional integrable soliton equations in the sense that the Dubrovin equations, combined with appropriate trace formulas, encode all hierarchies of soliton equations associated with hyperelliptic curves. In other words, completely integable hierarchies of soliton equations determine Dubrovin equations and associated trace formulas and, vice versa, Dubrovin-type equations combined with trace formulas permit the construction of hierarchies of soliton equations.