A dispersive extension result for a class of nonlocal nonlinearities

Bastian Hilder; Christian Kuehn

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arXiv · 2609.05071

A dispersive extension result for a class of nonlocal nonlinearities

Bastian Hilder·Christian Kuehn

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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 Schr\"odinger 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 Schr\"odinger operator $\mathrm{i} \partial_x^2$. 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.

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Bastian Hilder, Christian Kuehn. 2026-09-04. A dispersive extension result for a class of nonlocal nonlinearities. https://arxiv.org/abs/2609.05071

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