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

Yvonne Alama Bronsard, Thierry Laurens

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.

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

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 . 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 merely belonging to for (BO) and for (CCM), the latter of which is precisely the scaling-critical regularity. Our main theorem states that the -norm of the error goes to zero as the truncation parameters go to infinity, uniformly on any bounded time interval . 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.
Paper Structure (15 sections, 13 theorems, 108 equations, 1 figure)

This paper contains 15 sections, 13 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Explicit formulas
  3. Results
  4. Outline
  5. Schemes based on the explicit formulas
  6. Defining the class of schemes
  7. Structure-preserving properties and examples
  8. Computational cost
  9. Fourier transforms, norms and resolvants
  10. Proof of convergence
  11. Operator bounds
  12. Self-adjoint operators and equivalence of norms
  13. Norm resolvent convergence
  14. The generated unitary groups
  15. Proof of Theorem

Key Result

Theorem 1.3

Let $u(t)$ denote the global solution to the eq:BO or eq:CS equation with initial data $u_0$ in $L^2(\mathbb{T})$ or $L^2_+(\mathbb{T})$ respectively, with $\| u_0 \|_{L^2} < 1$ in the case of the focusing eq:CS equation. Denote by $u_K(t)$ the numerical scheme eq:newScheme corresponding to $u_0$ wi

Figures (1)

  • Figure 1: Evolution of the \ref{['eq:BO']} equation from the step-function initial data $u_0(x)=\operatorname{sgn}(x)$. The first three panels illustrate the Talbot effect at the rational times $t=\pi/2$, $t=\pi/3$, and $t=\pi/6$, while the last panel corresponds to the irrational time $t=\sqrt{2}\,\pi$. The scheme \ref{['eq:linscheme']}, shown in green, approximates the linearized equation. The scheme \ref{['eq:newScheme BO']}, shown in blue, uses the spectral truncation $n(k)=K/2$ for $0\le k\le K/2$ and $n(k)=0$ otherwise and approximates \ref{['eq:BO']}. We take $K=2^{10}$.

Theorems & Definitions (36)

  • Remark 1.1
  • Remark 1.2: The cubic Szegő equation
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5: Convergence of $u_K$ for higher regularity data
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • ...and 26 more