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arXiv · 2609.27190

The Fermi-Pasta-Ulam system and the Korteweg-de Vries equation: a low-regularity continuum limit

Herbert Koch·Ruoyuan Liu
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Abstract

For the infinite Fermi-Pasta-Ulam (FPU) system, general solutions can be approximated by counter-propagating waves associated with solutions to the Korteweg-de Vries (KdV) equation as the lattice mesh size goes to zero. We show that, by exploiting the conservation of the FPU Hamiltonian, the continuum limit from the FPU system to the KdV equation with $L^2$-level initial data holds in appropriate norms on an arbitrary time interval, thereby answering an open question posed by Hong, Kwak, and Yang (2021). Moreover, for the local-in-time continuum limit of the FPU system to the KdV equation, we lower the required Sobolev regularity to $H^s$ with $s > - \frac 34$. Our continuum limit results also apply to the case of the Toda lattice in Flaschka's form, thereby lowering the regularity requirement in our previous work (2026). To establish this low-regularity continuum limit, we prove key trilinear estimates that are sharp up to the endpoint by combining linear estimates, the transversality of characteristic curves, and multilinear dispersive smoothing properties of the linear FPU flow.

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Herbert Koch, Ruoyuan Liu. 2026-09-23. The Fermi-Pasta-Ulam system and the Korteweg-de Vries equation: a low-regularity continuum limit. https://arxiv.org/abs/2609.27190

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