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, $H_V=-\partial_x^2+V(x)$, 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 $L^2(\mathbb R)$ 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.