Gauge transform for the Korteweg-de Vries equation and well-posedness below the $H^{-1}$-scale
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| author | Chapouto, Andreia Correia, Simão Ramos, João Pedro |
| author_facet | Chapouto, Andreia Correia, Simão Ramos, João Pedro |
| contents | We propose a new formulation of the Korteweg-de Vries equation (KdV) on the real line, via a gauge transform. While KdV and the gauged equation are equivalent for smooth solutions, the latter is better behaved at low regularity in Fourier-Lebesgue spaces. In particular, the admissible regularities go beyond the $H^{-1}$-scale, which is a well-known threshold for KdV. As a byproduct, by reversing the gauge transform, we are able to improve on the known theory for KdV and derive sharp local well-posedness in Fourier-Lebesgue spaces with large integrability exponent. Our strategy is based on an infinite normal form reduction and Fourier restriction estimates, together with a thorough exploitation of algebraic cancellations. Additionally, our method is totally independent of the KdV completely integrable structure, and extends to other non-integrable models with quadratic nonlinearities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15060 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gauge transform for the Korteweg-de Vries equation and well-posedness below the $H^{-1}$-scale Chapouto, Andreia Correia, Simão Ramos, João Pedro Analysis of PDEs We propose a new formulation of the Korteweg-de Vries equation (KdV) on the real line, via a gauge transform. While KdV and the gauged equation are equivalent for smooth solutions, the latter is better behaved at low regularity in Fourier-Lebesgue spaces. In particular, the admissible regularities go beyond the $H^{-1}$-scale, which is a well-known threshold for KdV. As a byproduct, by reversing the gauge transform, we are able to improve on the known theory for KdV and derive sharp local well-posedness in Fourier-Lebesgue spaces with large integrability exponent. Our strategy is based on an infinite normal form reduction and Fourier restriction estimates, together with a thorough exploitation of algebraic cancellations. Additionally, our method is totally independent of the KdV completely integrable structure, and extends to other non-integrable models with quadratic nonlinearities. |
| title | Gauge transform for the Korteweg-de Vries equation and well-posedness below the $H^{-1}$-scale |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.15060 |