Infinite-Length Limit of Spectral Curves and Inverse Scattering

TL;DR

This paper explores the relationship between spectral curves and inverse scattering in classical integrable models, demonstrating how spectral data transforms into scattering data as the system length approaches infinity, with examples from KdV and Heisenberg models.

Contribution

It provides a detailed analysis of the infinite-length limit of spectral curves and their connection to inverse scattering data in classical integrable field theories.

Findings

Spectral curves degenerate into solitons at infinite length.

Transformation of spectral data into scattering data is illustrated for KdV and Heisenberg models.

The relationship between periodic spectral data and asymptotic scattering data is clarified.

Abstract

Integrability equips models of theoretical physics with efficient methods for the exact construction of useful states and their evolution. Relevant tools for classical integrable field models in one spatial dimensional are spectral curves in the case of periodic fields and inverse scattering for asymptotic boundary conditions. Even though the two methods are quite different in many ways, they ought to be related by taking the periodicity length of closed boundary conditions to infinity. Using the Korteweg-de Vries equation and the continuous Heisenberg magnet as prototypical classical integrable field models, we discuss and illustrate how data for spectral curves transforms into asymptotic scattering data. In order to gain intuition and also for concreteness, we review how the elliptic states of these models degenerate into solitons at infinite length.