Lagrangian invariant tori for completely resonant nonlinear Schrödinger equations
Abstract
We study small solutions of the parameterless nonlinear Schrödinger equations on the circle. In this regime, the nonlinearity is perturbative and the system is thus nearly integrable. We prove the persistence, in the presence of nonlinearity, of full-dimensional tori. We justify this denomination by proving that they are Lagrangian in the sense that, although they are not submanifolds (since they are compact), at every point their tangent cone is a Lagrangian subspace. This is the first result of this type for an equation without an infinity of external random parameters. We achieve this by establishing new fine decay and uniformity properties of the amplitude-to-frequency map. This is the main technical novelty of this paper. It allows us to modulate all the frequencies and thus obtain non-resonance conditions without any assumption on the choice of the tangential sites.
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BibTeX
@misc{bernier2026lagrangian,
title = {{Lagrangian invariant tori for completely resonant nonlinear Schrödinger equations}},
author = {Joackim Bernier and Michela Procesi},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26044},
eprint = {2610.26044},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05802370v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lagrangian invariant tori for completely resonant nonlinear Schrödinger equations
AU - Joackim Bernier
AU - Michela Procesi
PY - 2026
DA - 2026/10/16
PB - HAL
DO - 10.48550/arxiv.2610.26044
UR - https://hal.science/hal-05802370v1
AB - We study small solutions of the parameterless nonlinear Schrödinger equations on the circle. In this regime, the nonlinearity is perturbative and the system is thus nearly integrable. We prove the persistence, in the presence of nonlinearity, of full-dimensional tori. We justify this denomination by proving that they are Lagrangian in the sense that, although they are not submanifolds (since they are compact), at every point their tangent cone is a Lagrangian subspace. This is the first result of this type for an equation without an infinity of external random parameters. We achieve this by establishing new fine decay and uniformity properties of the amplitude-to-frequency map. This is the main technical novelty of this paper. It allows us to modulate all the frequencies and thus obtain non-resonance conditions without any assumption on the choice of the tangential sites.
ER -