发表机构
Columbia University（哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对紧支撑势的聚焦一维三次NLS-GP方程，建立了共振诱导非线性束缚态在分叉点附近的轨道稳定性与不稳定性精确判据，并指出在共振频率足够小且本征模严格正时态是稳定的，辅以数值模拟验证。
AI 中文摘要
我们研究聚焦一维三次非线性薛定谔/ Gross-Pitaevskii 方程（NLS-GP），其中底层线性薛定谔算子 $H_V=-\partial_x^2+V(x)$ 的势函数具有紧支撑。在文献 \cite{turner2026resonance} 中，作者证明了 $H_V$ 的反射系数的纯虚零点（透射共振）或下半平面中的纯虚极点（散射共振）会生成共振诱导非线性束缚态的分支，并且这些态在严格正的 $L^2(\mathbb R)$ 激发阈值处分叉。这与从 $H_V$ 的点谱（对应于上半平面中的极点）在零 $L^2$ 范数处分叉的非线性束缚态形成对比。在本文中，我们建立了分叉点附近共振诱导态的轨道稳定性与不稳定性的精确判据。一个推论是，如果 i) 底层线性（散射或透射）共振频率足够小，且 ii) 相应的线性共振本征模严格为正，则共振诱导非线性态是轨道稳定的。我们进行了数值模拟，以探索我们的理论无法触及的参数区域，并研究初始条件接近稳定和不稳定共振诱导非线性束缚态时的大时间动力学。
英文摘要
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.
Comments45 pages, 12 figures
