High-low pressure domain wall for the classical Toda lattice

TL;DR

This paper investigates the classical Toda lattice with domain wall initial conditions, analyzing the hydrodynamic behavior and demonstrating smooth self-similar solutions despite apparent singularities.

Contribution

It provides numerical solutions showing that the hydrodynamic equations for the Toda lattice are smooth at zero stretch, resolving apparent singularities.

Findings

Self-similar solutions are smooth at zero stretch.

Hydrodynamic profiles scale ballistically in the domain wall setup.

Numerical evidence supports the regularity of solutions despite singular equations.

Abstract

We study the classical Toda lattice with domain wall initial conditions, for which left and right half lattice are in thermal equilibrium but with distinct parameters of pressure, mean velocity, and temperature. In the hydrodynamic regime the respective space-time profiles scale ballisticly. The particular case of interest is a jump from low to high pressure at uniform temperature and zero mean velocity. Thereby the scaling function for the average stretch (also free volume) is forced to change sign. By direct inspection, the hydrodynamic equations for the Toda lattice seem to be singular at zero stretch. In our contribution we report on numerical solutions and convincingly establish that nevertheless the self-similar solution exhibits smooth behavior.