Fourier Energy Strata for the Cubic Schrödinger Equation
Abstract
We prove a geometric hierarchy of Fourier layer energies for the weakly nonlinear cubic Schrodinger equation on 𝕋π×𝕋1, retaining its nontrivial resonant dynamics. For initial Fourier support in a rectangle, form nested rectangles with half-widths (2j+1) times the initial ones; layers are the differences between consecutive rectangles. On 0≤ t≤ Tδ−1, the energy in layer j, measured by its squared H1 norm, is at most CNδ2j for 1≤ j≤ N; the energy outside the Nth rectangle is at most CNδ2N+2. These bounds hold at every fixed depth N under a uniform Hs bound, s>2, with constants depending on N. The Sobolev bound holds for every fixed T in the defocusing case. We also obtain layer estimates for support bounded in one coordinate and for geometrically decaying initial Fourier tails. The proof uses a filtered Poincare-Dulac normal form: input layers select which nonresonant interactions are removed, while the same layer weights control the interactions left in the equation. Finite coordinate changes and stability estimates transfer this hierarchy to the exact solution.
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BibTeX
@misc{xie2026fourier,
title = {{Fourier Energy Strata for the Cubic Schrödinger Equation}},
author = {Yulin Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05165},
eprint = {2610.05165},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05165},
}
RIS
TY - UNPB
TI - Fourier Energy Strata for the Cubic Schrödinger Equation
AU - Yulin Xie
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05165
UR - https://arxiv.org/abs/2610.05165
AB - We prove a geometric hierarchy of Fourier layer energies for the weakly nonlinear cubic Schrodinger equation on 𝕋_π×𝕋_1, retaining its nontrivial resonant dynamics. For initial Fourier support in a rectangle, form nested rectangles with half-widths (2j+1) times the initial ones; layers are the differences between consecutive rectangles. On 0≤ t≤ Tδ^−1, the energy in layer j, measured by its squared H^1 norm, is at most C_Nδ^2j for 1≤ j≤ N; the energy outside the Nth rectangle is at most C_Nδ^2N+2. These bounds hold at every fixed depth N under a uniform H^s bound, s>2, with constants depending on N. The Sobolev bound holds for every fixed T in the defocusing case. We also obtain layer estimates for support bounded in one coordinate and for geometrically decaying initial Fourier tails. The proof uses a filtered Poincare-Dulac normal form: input layers select which nonresonant interactions are removed, while the same layer weights control the interactions left in the equation. Finite coordinate changes and stability estimates transfer this hierarchy to the exact solution.
ER -