The Whitham Approach to the $c\to0$ limit of The Lieb-Liniger Model and Generalized Hydrodynamics

TL;DR

This paper demonstrates that the Whitham approach is the semiclassical limit of generalized hydrodynamics for the Lieb-Liniger model as the interaction parameter approaches zero, linking quantum and classical descriptions of integrable systems.

Contribution

It explicitly connects the Whitham and generalized hydrodynamics methods, showing the former as the semiclassical limit of the latter in the Lieb-Liniger model.

Findings

The Whitham approach corresponds to the semiclassical limit of generalized hydrodynamics.

Quantum expectation values can be computed in the $c o0$ limit using this connection.

The work bridges classical and quantum integrable systems through explicit formalism.

Abstract

The Whitham approach is a well-studied method to describe non-linear integrable systems. Although approximate in nature, its results may predict rather accurately the time evolution of such systems in many situations given initial conditions. A similarly powerful approach has recently emerged that is applicable to quantum integrable systems, namely the generalized hydrodynamics approach. This paper aims at showing that the Whitham approach is the semiclassical limit of the generalized hydrodynamics approach by connecting the two formal methods explicitly on the example of the Lieb-Liniger model on the quantum side to the non-linear Schr\"{o}dinger equation on the classical side in the $c\to0$ limit, $c$ being the interaction parameter. We show how quantum expectation values may be computed in this limit based on the connection established here which is mentioned above.