Uniform-in-Time Approximation of Calogero-Moser Particles by Continuum Multisolitons
Abstract
We prove that every solution of the classical rational Calogero-Moser system can be approximated, after subtracting a common linear drift from the particle positions, by rational multisoliton solutions of the focusing continuum Calogero-Moser equation, uniformly for all time. The construction identifies the multisoliton inverse spectral matrix as a rank-one dissipative perturbation of Moser's time-dependent Hermitian position matrix. A uniform lower bound on the particle separation then yields quantitative control of the multisoliton poles. Their real parts and real velocities approximate the shifted particle positions and velocities with errors of order ε2, while the pole heights are positive and sum exactly to ε. Using the exact Poisson-kernel representation of the multisoliton density, we obtain convergence to the atomic particle measure in bounded-Lipschitz distance at rate O(ε(1+|logε|)), uniformly in time. We also derive the large-time distribution of pole height at fixed ε, with one distinguished branch retaining the limiting height and the remaining heights decaying quadratically in time. Explicit two-particle formulas illustrate the approximation and show that the real-position error estimate is sharp.
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BibTeX
@misc{wang2026uniformintime,
title = {{Uniform-in-Time Approximation of Calogero-Moser Particles by Continuum Multisolitons}},
author = {Hedong Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21314},
eprint = {2609.21314},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21314},
}
RIS
TY - UNPB
TI - Uniform-in-Time Approximation of Calogero-Moser Particles by Continuum Multisolitons
AU - Hedong Wang
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21314
UR - https://arxiv.org/abs/2609.21314
AB - We prove that every solution of the classical rational Calogero-Moser system can be approximated, after subtracting a common linear drift from the particle positions, by rational multisoliton solutions of the focusing continuum Calogero-Moser equation, uniformly for all time. The construction identifies the multisoliton inverse spectral matrix as a rank-one dissipative perturbation of Moser's time-dependent Hermitian position matrix. A uniform lower bound on the particle separation then yields quantitative control of the multisoliton poles. Their real parts and real velocities approximate the shifted particle positions and velocities with errors of order ε^2, while the pole heights are positive and sum exactly to ε. Using the exact Poisson-kernel representation of the multisoliton density, we obtain convergence to the atomic particle measure in bounded-Lipschitz distance at rate O(ε(1+|logε|)), uniformly in time. We also derive the large-time distribution of pole height at fixed ε, with one distinguished branch retaining the limiting height and the remaining heights decaying quadratically in time. Explicit two-particle formulas illustrate the approximation and show that the real-position error estimate is sharp.
ER -