Infinite-Length Limit of Spectral Curves and Inverse Scattering

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.