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The Korteweg-de Vries limit for the global dynamics of the Toda lattice
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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BibTeX
@misc{koch2026kortewegde,
title = {{The Korteweg-de Vries limit for the global dynamics of the Toda lattice}},
author = {Herbert Koch and Ruoyuan Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.07532},
eprint = {2605.07532},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.07532},
}
RIS
TY - UNPB
TI - The Korteweg-de Vries limit for the global dynamics of the Toda lattice
AU - Herbert Koch
AU - Ruoyuan Liu
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2605.07532
UR - https://arxiv.org/abs/2605.07532
AB - 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.
ER -