The Korteweg-de Vries limit for the global dynamics of the Toda lattice
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911662068465664 |
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| author | Liu, Ruoyuan Koch, Herbert |
| author_facet | Liu, Ruoyuan Koch, Herbert |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07532 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Korteweg-de Vries limit for the global dynamics of the Toda lattice Liu, Ruoyuan Koch, Herbert Exactly Solvable and Integrable Systems Analysis of PDEs 35Q53, 37J70, 37K10, 37K60 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. |
| title | The Korteweg-de Vries limit for the global dynamics of the Toda lattice |
| topic | Exactly Solvable and Integrable Systems Analysis of PDEs 35Q53, 37J70, 37K10, 37K60 |
| url | https://arxiv.org/abs/2605.07532 |