On the convergence of explicit formulas for $L^2$ solutions to the Benjamin-Ono and continuum Calogero-Moser equations

TL;DR

This paper develops and analyzes fully-discrete schemes for the Benjamin-Ono and continuum Calogero-Moser equations, proving convergence for rough solutions and demonstrating the Talbot effect numerically.

Contribution

It introduces structure-preserving, fully-discrete schemes for BO and CCM equations with proven convergence for low regularity solutions.

Findings

Convergence of schemes for $L^2$ solutions established.

Numerical evidence of the Talbot effect for BO.

Schemes preserve mass and momentum for BO.

Abstract

By developing discrete counterparts to recent advances in nonlinear integrability, and in particular to the discovery of explicit formulas, we design and analyze fully-discrete approximations to the Benjamin-Ono (BO) and continuum Calogero-Moser (CCM) equations on the torus. We build on the key observation that discretizing such explicit formulas yields schemes that are exact in time (requiring only spatial discretization) and have a computational cost independent of the final time $T$. In this work, we first generalize the fully-discrete schemes of arXiv:2412.13480 to include numerical approximations with better structure preservation properties, including the conservation of mass and momentum in the case of the (BO) equation. Secondly, building on recent analyses of the corresponding Lax operators, we extend the convergence results to this class of schemes for rough solutions $u(t)$ merely belonging to $L^2(\mathbb{T})$ for (BO) and $L^2_{+}(\mathbb{T})$ for (CCM), the latter of which is precisely the scaling-critical regularity. Our main theorem states that the $L^2(\mathbb{T})$-norm of the error goes to zero as the truncation parameters go to infinity, uniformly on any bounded time interval $[-T,T]$. As an example, we apply our scheme to the (BO) equation with a square-wave initial profile, and obtain the first numerical evidence of the Talbot effect for (BO) supported by a rigorous convergence result.