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arXiv · 2605.07532

The Korteweg-de Vries limit for the global dynamics of the Toda lattice

Ruoyuan Liu·Herbert Koch
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Abstract

It has been observed that the dynamics of the Toda lattice can be well described by solutions of the Korteweg-de Vries (KdV) equation in the continuum limit. We show that, under the KdV scaling and a suitable translation, the solution of the Toda lattice with H^1 initial data converges to that of the KdV equation globally in time. Our proof relies on tools from harmonic analysis and also on the construction and the conservation of mass and energy of the Toda lattice, the latter of which are derived from the completely integrable structure of the Toda lattice. As a consequence, we obtain long-wave KdV limits for the Toda lattice.

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Ruoyuan Liu, Herbert Koch. 2026-05-08. The Korteweg-de Vries limit for the global dynamics of the Toda lattice. https://arxiv.org/abs/2605.07532

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