The Whitham Approach to the $c\to 0$ limit of The Lieb-Liniger Model and Generalized Hydrodynamics

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 Schrödinger equation on the classical side in the $c\to 0$ 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.

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• Lecture notes on Generalised Hydrodynamics
• Generalized Hydrodynamics in the 1D Bose gas: theory and experiments
• Few-body Bose gases in low dimensions -- a laboratory for quantum dynamics
• Generalized hydrodynamics of the KdV soliton gas
• Generalized Hydrodynamics of the attractive Non-Linear Schroedinger Equation
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• Probing the local rapidity distribution of a 1D Bose gas
• The Whitham approach to Generalized Hydrodynamics
• Benchmarks of Generalized Hydrodynamics for 1D Bose Gases
• Characterising transport in a quantum gas by measuring Drude weights
• Viscous Flow in a 1D Spin-Polarized Fermi Gas: the Role of Integrability on Viscosity
• Quantum fluctuating theory for one-dimensional shock waves
• Exact Matrix Elements of the Field Operator in the Thermodynamic Limit of the Lieb-Liniger Model