Almost-sure quenched KAM tori for cubic NLS with a spatial white-noise potential
Abstract
Let Aω=−∂x2+ρḂx(ω), ρ≠0, be the Dirichlet Schrödinger operator on (0,π) with a spatial Gaussian white-noise potential, realized pathwise via quasi-derivatives. Here Ḃx denotes the distributional derivative of Brownian motion with respect to the spatial variable x. For κ≠0, we consider the cubic nonlinear Schrödinger equation i ut=Aωu+κ|u|2u. We first prove a zero-set theorem for locally real-analytic functions on classical Wiener space. As a consequence, on a single event of probability one, no nontrivial finitely supported integer combination of the random eigenvalues vanishes, and the quartic twist matrix is nonsingular for every finite tangential set. After fixing a path in this event, we combine these qualitative nondegeneracy properties with a partial quartic Birkhoff normal form adapted to the KAM scheme. For every finite nonempty tangential set J of cardinality b, we obtain a Cantor family of linearly stable, real-analytic, small-amplitude invariant b-tori. The family is parametrized by a Cantor subset of [ν,2ν]b whose relative measure tends to one as ν→0.
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BibTeX
@misc{dong2026almostsure,
title = {{Almost-sure quenched KAM tori for cubic NLS with a spatial white-noise potential}},
author = {Yingdu Dong and Wenwen Jian and Xiaoping Yuan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22808},
eprint = {2609.22808},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22808},
}
RIS
TY - UNPB
TI - Almost-sure quenched KAM tori for cubic NLS with a spatial white-noise potential
AU - Yingdu Dong
AU - Wenwen Jian
AU - Xiaoping Yuan
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22808
UR - https://arxiv.org/abs/2609.22808
AB - Let A_ω=−∂_x^2+ρḂ_x(ω), ρ≠0, be the Dirichlet Schrödinger operator on (0,π) with a spatial Gaussian white-noise potential, realized pathwise via quasi-derivatives. Here Ḃ_x denotes the distributional derivative of Brownian motion with respect to the spatial variable x. For κ≠0, we consider the cubic nonlinear Schrödinger equation i u_t=A_ωu+κ|u|^2u. We first prove a zero-set theorem for locally real-analytic functions on classical Wiener space. As a consequence, on a single event of probability one, no nontrivial finitely supported integer combination of the random eigenvalues vanishes, and the quartic twist matrix is nonsingular for every finite tangential set. After fixing a path in this event, we combine these qualitative nondegeneracy properties with a partial quartic Birkhoff normal form adapted to the KAM scheme. For every finite nonempty tangential set J of cardinality b, we obtain a Cantor family of linearly stable, real-analytic, small-amplitude invariant b-tori. The family is parametrized by a Cantor subset of [ν,2ν]^b whose relative measure tends to one as ν→0.
ER -