Gauge transform for the Korteweg-de Vries equation and well-posedness below the H−1-SCALE
Abstract
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.
Related
- Normal form approach to unconditional well-posedness of the periodic stochastic Korteweg-de Vries equation with an additive noise
- Long-time validity of the Korteweg-de Vries approximation for the Boussinesq equation
- Convergence Rate toward Rarefaction Wave under Periodic Perturbation for Generalized Korteweg-de Vries-Burgers equation
- The Fermi-Pasta-Ulam system and the Korteweg-de Vries equation: a low-regularity continuum limit
- Asymptotic preserving IMEX Active Flux schemes for a hyperbolic approximation of the Korteweg-de Vries equation
BibTeX
@misc{chapouto2026gauge,
title = {{Gauge transform for the Korteweg-de Vries equation and well-posedness below the H−1-SCALE}},
author = {Chapouto, Andreia and Correia, Simão and Ramos, João Pedro},
howpublished = {HAL},
year = {2026},
month = aug,
eprint = {2601.15060},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05851594v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gauge transform for the Korteweg-de Vries equation and well-posedness below the H−1-SCALE
AU - Chapouto, Andreia
AU - Correia, Simão
AU - Ramos, João Pedro
PY - 2026
DA - 2026/08/25
PB - HAL
UR - https://hal.science/hal-05851594v1
AB - 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.
ER -