A new quadrature for the generalized hydrodynamics equation and absence of shocks in the Lieb-Liniger model

TL;DR

This paper introduces a new numerical method for solving generalized hydrodynamics equations in the Lieb-Liniger model, demonstrating the absence of shocks and providing new insights into non-equilibrium correlations in integrable quantum gases.

Contribution

It presents a novel quadrature solution for GHD equations, showing that shocks do not form in the Lieb-Liniger model and enabling new correlation calculations in non-stationary states.

Findings

No shock formation in Lieb-Liniger model confirmed

New quadrature method improves GHD numerical solutions

Derived correlation expressions for non-equilibrium states

Abstract

In conventional fluids, it is well known that Euler-scale equations are plagued by ambiguities and instabilities. Smooth initial conditions may develop shocks, and weak solutions, such as for domain wall initial conditions (the paradigmatic Riemann problem of hydrodynamics), are not unique. The absence of shock formation experimentally observed in quasi-one-dimensional cold-atomic gases, which are described by the Lieb-Liniger model, provides perhaps the strongest pointer to a modification of the hydrodynamic equation due to integrability. Generalised hydrodynamics (GHD) is the required hydrodynamic theory, taking into account the infinite number of conserved quantities afforded by integrability. We provide a new quadrature for the GHD equation -- a solution in terms of a Banach fixed-point problem where time has been explicitly integrated. The quadrature is an efficient numerical solution tool; and it allows us, in the Lieb-Liniger model, to rigorously show that no shock may appear at all times, and, when combined with recent hydrodynamic fluctuation theories, to obtain new expressions for correlations in non-stationary states, establishing for the first time the presence of discontinuities characteristic of the non-equilibrium dynamics.