PreprintarXiv licencearXiv
A dispersive extension result for a class of nonlocal nonlinearities
Abstract
We prove that a class of nonlocal nonlinearities on periodic Sobolev spaces can be expressed as a trace of the unique mild solution to a local dispersive problem consisting of a system of forced Schrödinger equations. The proof is based on a reformulation of the nonlocal nonlinearities as the resonant orbit average of a real-analytic function under the unitary group generated by the free Schrödinger operator i ∂x2. We illustrate our result by providing an equivalent local reformulation for a recently obtained nonlocal amplitude equation that formally captures the dynamics of parabolic systems close to a conserved Hopf instability.
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BibTeX
@misc{hilder2026dispersive,
title = {{A dispersive extension result for a class of nonlocal nonlinearities}},
author = {Bastian Hilder and Christian Kuehn},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05071},
eprint = {2609.05071},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05071},
}
RIS
TY - UNPB
TI - A dispersive extension result for a class of nonlocal nonlinearities
AU - Bastian Hilder
AU - Christian Kuehn
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05071
UR - https://arxiv.org/abs/2609.05071
AB - We prove that a class of nonlocal nonlinearities on periodic Sobolev spaces can be expressed as a trace of the unique mild solution to a local dispersive problem consisting of a system of forced Schrödinger equations. The proof is based on a reformulation of the nonlocal nonlinearities as the resonant orbit average of a real-analytic function under the unitary group generated by the free Schrödinger operator i ∂_x^2. We illustrate our result by providing an equivalent local reformulation for a recently obtained nonlocal amplitude equation that formally captures the dynamics of parabolic systems close to a conserved Hopf instability.
ER -