2024/06/26 by Friedrich Hübner, Benjamin Doyon, Hübner, Friedrich +1 · 3 citations
Engineering · Mathematics · Physics and Astronomy · #Computational Fluid Dynamics and Aerodynamics #FOS: Physical sciences #Nonlinear Waves and Solitons #Numerical methods for differential equations #Statistical Mechanics (cond-mat.stat-mech)
paper · pdf · doi:10.48550/arxiv.2406.18322
openalex publication_date 2024/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.