Uniform-in-Time Approximation of Calogero-Moser Particles by Continuum Multisolitons

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 $\varepsilon^2$, while the pole heights are positive and sum exactly to $\varepsilon$. Using the exact Poisson-kernel representation of the multisoliton density, we obtain convergence to the atomic particle measure in bounded-Lipschitz distance at rate $O(\varepsilon(1+|\log\varepsilon|))$, uniformly in time. We also derive the large-time distribution of pole height at fixed $\varepsilon$, 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.