Weakly turbulent solution to the Schrödinger equation on the two-dimensional torus with real potential decaying to zero at infinity

Chabert, Ambre

doi:10.2140/apde.2025.18.2061

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Weakly turbulent solution to the Schrödinger equation on the two-dimensional torus with real potential decaying to zero at infinity

Ambre Chabert

Vol. 18 (2025), No. 8, 2061–2080

DOI: 10.2140/apde.2025.18.2061

Abstract

We build a smooth time-dependent real potential on the two-dimensional torus, decaying as time tends to infinity in Sobolev norms along with all its time derivatives, and we exhibit a smooth solution to the associated Schrödinger equation on the two-dimensional torus whose $H^{s}$ norms nevertheless grow logarithmically as time tends to infinity. We use Fourier decomposition in order to exhibit a discrete resonant system of interactions, which we are further able to reduce to a sequence of finite-dimensional linear systems along which the energy propagates to higher and higher frequencies. The constructions are very explicit, and we can thus obtain lower bounds on the growth rate of the solution.

Keywords

linear Schrödinger equation, weak turbulence, resonant system, forward cascade of energy, backward integration

Mathematical Subject Classification

Primary: 35B40

Secondary: 35Q41

Milestones

Received: 10 April 2024

Revised: 14 August 2024

Accepted: 20 September 2024

Published: 25 July 2025

Authors

Ambre Chabert
Département de Mathématiques et Applications Ecole Normale Supérieure Paris France

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Vol. 18, No. 8, 2025