Resonance-induced nonlinear bound states

TL;DR

This work shows that nonlinear bound states in the 1D focusing NLS/GP with a compactly supported potential can bifurcate not only from $L^2$ bound states but also from linear scattering features, including scattering resonance poles and transmission resonances of $H_V=-\partial_x^2+V$. The authors develop a rigorous bifurcation framework using an inner-outer construction and the implicit-function theorem to produce analytic energy branches $E(\varepsilon)$ emanating from simple purely imaginary linear modes $k_\star=i\kappa_\star$, with the bound-state solutions extending to full $L^2(\mathbb{R})$ profiles after suitable translations. A striking outcome is the appearance of a strictly positive $L^2$ threshold $\mathcal{N}_{\mathrm{thr}}>0$ for resonance- and transmission-induced bifurcations, in contrast to bound-state seeds which can have zero threshold; this is illustrated in a delta-potential example where a barrier yields $\mathcal{N}_{\mathrm{thr}}=8\sqrt{-E_\star}$ while a well does not. The paper also analyzes zero-energy threshold resonances, showing symmetric bifurcating branches with finite centerings $x_{\rm L}(\varepsilon),x_{\rm R}(\varepsilon)$ and providing explicit leading-order formulas. Overall, the results extend classical bifurcation theory for nonlinear bound states to resonance seeds, revealing rich structure in the $E$–$\mathcal{N}$ plane and guiding future explorations of stability, higher dimensions, and time-dependent scattering phenomena.

Abstract

We study nonlinear bound states -- time-harmonic and spatially decaying ($L^2$) solutions -- of the nonlinear Schrödinger / Gross--Pitaevskii equations (NLS/GP) with a compactly supported linear potential. Such solutions are known to bifurcate from the $L^2$ bound states of an underlying Schrödinger operator $H_V=-\partial_x^2+V$. In this article we prove an extension of this result: for the 1D NLS/GP, nonlinear bound states also arise via bifurcation from the scattering resonance states and transmission resonance states of $H_V$, associated with the poles and zeros, respectively, of the reflection coefficients, $r_\pm(k)$, of $H_V$. The corresponding resonance states are non-decaying and only $L^2_{\rm loc}$. In contrast to nonlinear states arising from $L^2$ bound states of $H_V$, these resonance bifurcations initiate at a strictly positive $L^2$ threshold which is determined by the position of the complex scattering resonance pole or transmission resonance zero.

Figures (10)

• Figure 1.1: Profiles $\psi(x;E,\alpha)$ for $V=\alpha\delta(x)$ (left) and norms $\mathcal{N}[\psi_E]$ (right). Top: potential barrier ($\alpha>0$), with solutions shaded from light to deep red as $E\to E_\star$; the profiles approach the threshold logarithmically, producing constant spatial shifts. Bottom: potential well ($\alpha<0$), shaded from light to deep cyan. Right: the $(E,\mathcal{N})$ branches terminate at $E_\star$, with an excitation threshold only for $\alpha>0$.
• Figure 3.1: Schematic illustration of Theorem \ref{['thm:nogo']} for nonlinearities $-|\psi|^{2\sigma}\psi$. The shaded grey areas denote forbidden regions in the $\bigl(\mathcal{N},E\bigr)$-plane. Left: subcritical and critical case $\sigma\le2$ when $H_V$ has no bound state or threshold resonance. Right: supercritical case $\sigma>2$ when $H_V$ has no threshold resonance. The symbols and mark branches bifurcating from scattering and transmission resonances of $H_V$ on $i\mathbb{R}$; see Theorem \ref{['thm:bifurcations']}. Each bifurcating branch emerges from the faint gray "free soliton guide curves" $E\mapsto \mathcal{N}[\mathcal{S}_{E,\sigma}]$ and $E\mapsto 2\,\mathcal{N}[\mathcal{S}_{E,\sigma}]$, where $\mathcal{S}_{E,\sigma}$ denotes the fundamental 1-soliton.
• Figure 4.1: The calculation \ref{['eq:xRL-zero']} for a square well potential $V(x) = -\frac{\pi^2}{4} \chi_{[-1,1]}(x)$ yields $x_{\rm R}(\varepsilon)\to 3/4$, for the odd nonlinear bound state (Dirichlet condition at $x=0$) arising from an odd threshold resonance mode $U_\star(x)$. Here, we verify this numerically by computing the solutions on a fine grid of $E \in [-10,0)$ and plotting $x_{\rm R}$ vs $E$.
• Figure 5.1: Schematic of bifurcation due to a bound state pole of $H_V$ at $k=i\kappa_b$, $\kappa_b>0$. Left panel: Highlighted is $k_b=i\kappa_b$ at which $w(k_b)=0$. The inset shows the corresponding anti-symmetric eigenstate $U_\star$ of $H_V$. Right panel: Curve of nonlinear bound states bifurcating from $\mathcal{N}=0$ at energy $E=E_b=k_b^2=-\kappa_b^2<0$.
• Figure 5.2: Schematic showing the bifurcation of a nonlinear bound state $(\psi_\star,k_{\rm r}^2)$ from a scattering resonance (anti-bound state) pole of $H_V$. Left panel: Highlighted is a resonance (anti-bound state) pole $k_r=i\kappa_r$, with $\kappa_r<0$, at which $w(k_r)=0$. The inset shows the corresponding anti-symmetric scattering resonance mode $U_\star$ of $H_V$. Right panel: Curve of nonlinear bound states bifurcating at a strictly positive $L^2$ threshold $\mathcal{N}=2\mathcal{N}[\mathcal{S}_{{k_r}^2}]>0$ at energy $E=k_r^2=-\kappa_r^2<0$.

Theorems & Definitions (13)

• Theorem 1.1: The 1-soliton of cubic NLS
• Theorem 3.1
• Remark 3.2: Generalization to power nonlinearities
• Lemma 3.3
• proof : Proof of Lemma \ref{['lem:bound-4']}
• proof : Proof of Theorem \ref{['thm:nogo']}
• Theorem 4.1: Bifurcations from Bound States and Scattering/Transmission Resonances
• Remark 4.2: Generalization to focusing nonlinearities
• Lemma 4.3
• proof