Stability and instability analysis of resonance-induced nonlinear bound states
Abstract
We study the focusing one-dimensional cubic nonlinear Schrödinger / Gross--Pitaevskii equation (NLS-GP), where the potential of the underlying linear Schrödinger operator, HV=−∂x2+V(x), is compactly supported. In \cite{turner2026resonance} the authors proved that purely imaginary zeros (transmission resonances) or purely imaginary poles in the lower half plane (scattering resonances) of a reflection coefficient of HV seed branches of {\it resonance-induced nonlinear bound states}, and that these states bifurcate at a strictly positive L2(ℝ) excitation threshold. This is in contrast to nonlinear bound states which bifurcate at zero L2-norm from point spectra of HV (corresponding to poles in the upper half plane). In this paper we establish precise criteria for the nonlinear orbital stability and instability of the resonance-induced states near the bifurcation point. A corollary is that resonance-induced nonlinear states are orbitally stable if i) the underlying linear (scattering or transmission) resonance frequency is sufficiently small, and ii) the corresponding linear resonance eigenmode is strictly positive. Numerical simulations are presented to explore regimes not accessible to our theory, and to explore the large time dynamics for initial conditions near stable and unstable resonance-induced nonlinear bound states.
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BibTeX
@misc{turner2026stability,
title = {{Stability and instability analysis of resonance-induced nonlinear bound states}},
author = {Jackson C. Turner and Michael I. Weinstein},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01875},
eprint = {2610.01875},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01875},
}
RIS
TY - UNPB
TI - Stability and instability analysis of resonance-induced nonlinear bound states
AU - Jackson C. Turner
AU - Michael I. Weinstein
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01875
UR - https://arxiv.org/abs/2610.01875
AB - We study the focusing one-dimensional cubic nonlinear Schrödinger / Gross--Pitaevskii equation (NLS-GP), where the potential of the underlying linear Schrödinger operator, H_V=−∂_x^2+V(x), is compactly supported. In \cite{turner2026resonance} the authors proved that purely imaginary zeros (transmission resonances) or purely imaginary poles in the lower half plane (scattering resonances) of a reflection coefficient of H_V seed branches of {\it resonance-induced nonlinear bound states}, and that these states bifurcate at a strictly positive L^2(ℝ) excitation threshold. This is in contrast to nonlinear bound states which bifurcate at zero L^2-norm from point spectra of H_V (corresponding to poles in the upper half plane). In this paper we establish precise criteria for the nonlinear orbital stability and instability of the resonance-induced states near the bifurcation point. A corollary is that resonance-induced nonlinear states are orbitally stable if i) the underlying linear (scattering or transmission) resonance frequency is sufficiently small, and ii) the corresponding linear resonance eigenmode is strictly positive. Numerical simulations are presented to explore regimes not accessible to our theory, and to explore the large time dynamics for initial conditions near stable and unstable resonance-induced nonlinear bound states.
ER -