Long time strong Sobolev instability for quasilinear NLS
Abstract
We prove long-time strong Sobolev instability for a class of quasilinear Schrödinger equations on 𝕋2 arising, via the Madelung transform, in the description of Euler-Korteweg capillary fluids. More precisely, for every s≥ 7, arbitrarily small μ>0, and arbitrarily large 𝒦>0, we construct a solution such that |u(0)|Hs<μ, |u(T)|Hs>𝒦 at some finite time T, for which we provide an explicit exponential upper bound. The instability mechanism is generated by a finite-dimensional resonant Toy Model associated with the cubic semilinear part of the equation, in the spirit of Colliander-Keel-Staffilani-Takaoka-Tao. The main technical difficulty is to justify this dynamics over the long time scale of the energy transfer in the presence of quasilinear derivative losses. We achieve this by combining a weak Birkhoff normal form with paradifferential calculus, microlocal symmetrization, and modified energy estimates. Our result provides a rigorous quasilinear realization of a resonant energy-transfer mechanism, and suggests a connection with weakly turbulent dynamics in capillary fluids.
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BibTeX
@misc{cacchio2026long,
title = {{Long time strong Sobolev instability for quasilinear NLS}},
author = {Yuri Cacchiò and Filippo Giuliani and Felice Iandoli and Raffaele Scandone},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37593},
eprint = {2609.37593},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37593},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Long time strong Sobolev instability for quasilinear NLS
AU - Yuri Cacchiò
AU - Filippo Giuliani
AU - Felice Iandoli
AU - Raffaele Scandone
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37593
UR - https://arxiv.org/abs/2609.37593
AB - We prove long-time strong Sobolev instability for a class of quasilinear Schrödinger equations on 𝕋^2 arising, via the Madelung transform, in the description of Euler-Korteweg capillary fluids. More precisely, for every s≥ 7, arbitrarily small μ>0, and arbitrarily large 𝒦>0, we construct a solution such that |u(0)|_H^s<μ, |u(T)|_H^s>𝒦 at some finite time T, for which we provide an explicit exponential upper bound. The instability mechanism is generated by a finite-dimensional resonant Toy Model associated with the cubic semilinear part of the equation, in the spirit of Colliander-Keel-Staffilani-Takaoka-Tao. The main technical difficulty is to justify this dynamics over the long time scale of the energy transfer in the presence of quasilinear derivative losses. We achieve this by combining a weak Birkhoff normal form with paradifferential calculus, microlocal symmetrization, and modified energy estimates. Our result provides a rigorous quasilinear realization of a resonant energy-transfer mechanism, and suggests a connection with weakly turbulent dynamics in capillary fluids.
ER -