The Quench Map in an Integrable Classical Field Theory: Nonlinear Schr\"odinger Equation

TL;DR

This paper investigates the non-equilibrium dynamics of the classical nonlinear Schr"odinger equation after a parameter quench, introducing the quench map on scattering data and exploring its properties and applications.

Contribution

It develops the concept of the quench map for integrable classical field theories, providing a detailed analysis and new methods for understanding parameter changes in the nonlinear Schr"odinger equation.

Findings

Explicit one-soliton solutions described by inverse scattering

Introduction of the quench map acting on scattering data

Application of Darboux-B"acklund transformations and Gelfand-Levitan-Marchenko equations

Abstract

We study the non-equilibrium dynamics obtained by an abrupt change (a {\em quench}) in the parameters of an integrable classical field theory, the nonlinear Schr\"odinger equation. We first consider explicit one-soliton examples, which we fully describe by solving the direct part of the inverse scattering problem. We then develop some aspects of the general theory using elements of the inverse scattering method. For this purpose, we introduce the {\em quench map} which acts on the space of scattering data and represents the change of parameter with fixed field configuration (initial condition). We describe some of its analytic properties by implementing a higher level version of the inverse scattering method, and we discuss the applications of Darboux-B\"acklund transformations, Gelfand-Levitan-Marchenko equations and the Rosales series solution to a related, dual quench problem. Finally, we comment on the interplay between quantum and classical tools around the theme of quenches and on the usefulness of the quantization of our classical approach to the quantum quench problem.