Asymptotic stability and sharp decay near homogeneous solutions of a nonlinearly damped Schrödinger equation on the torus
Abstract
We study the one-dimensional periodic nonlinear Schrödinger equation with a power-type nonlinear damping, both in the focusing and the defocusing regime. For a sufficiently small damping parameter and initial data close in relative H1 norm to a nonzero spatially homogeneous state, we prove global existence and asymptotic convergence, after a constant phase adjustment, to an explicit homogeneous solution. The L2 and H1 norms follow the explicit adiabatic decay law, with the same leading coefficient as the homogeneous solution. We obtain explicit convergence rates, including a t−1/2 bound for the relative spatial perturbation. For every nonconstant initial datum in a small neighborhood of the homogeneous solution, the kinetic energy has a strictly positive leading coefficient and decays one power of time faster than the squared L2 norm.
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BibTeX
@misc{shakarov2026asymptotic,
title = {{Asymptotic stability and sharp decay near homogeneous solutions of a nonlinearly damped Schrödinger equation on the torus}},
author = {Boris Shakarov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16526},
eprint = {2610.16526},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16526},
}
RIS
TY - UNPB
TI - Asymptotic stability and sharp decay near homogeneous solutions of a nonlinearly damped Schrödinger equation on the torus
AU - Boris Shakarov
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16526
UR - https://arxiv.org/abs/2610.16526
AB - We study the one-dimensional periodic nonlinear Schrödinger equation with a power-type nonlinear damping, both in the focusing and the defocusing regime. For a sufficiently small damping parameter and initial data close in relative H^1 norm to a nonzero spatially homogeneous state, we prove global existence and asymptotic convergence, after a constant phase adjustment, to an explicit homogeneous solution. The L^2 and H^1 norms follow the explicit adiabatic decay law, with the same leading coefficient as the homogeneous solution. We obtain explicit convergence rates, including a t^−1/2 bound for the relative spatial perturbation. For every nonconstant initial datum in a small neighborhood of the homogeneous solution, the kinetic energy has a strictly positive leading coefficient and decays one power of time faster than the squared L^2 norm.
ER -