Global attraction to the solitary manifold for the one-dimensional Schrödinger equation coupled to a nonlinear oscillator

Buslaev, Komech, Kopylova and Stuart (Comm. PDE 2008) proved that solutions of the one-dimensional Schrödinger equation coupled to a nonlinear oscillator — a nonlinear Schrödinger equation whose nonlinearity is concentrated at a single point via a Dirac delta — converge, for initial data close to a linearly stable solitary wave and under a spectral condition excluding internal modes, to a solitary wave plus dispersive radiation, with remainder decaying like $|t|^{-1/2}$. The authors explicitly left two problems open: (i) whether these soliton-plus-radiation asymptotics hold for every finite-energy solution, i.e. global attraction to the solitary manifold; and (ii) the extension to the parameter regime in which the linearization possesses an oscillatory internal mode. Problem (ii) was substantially resolved by Komech, Kopylova and Stuart (CPAA 2012) for the resonant sub-range of the internal-mode regime under a generic Fermi Golden Rule condition, leaving only the non-resonant sub-range untreated. Problem (i) — global attraction for a nonlinear Schrödinger equation with point-concentrated nonlinearity — remains fully open: the corresponding Klein–Gordon, wave, and Dirac results of the Komech program do not transfer to Schrödinger dispersion.

Background

The nonlinear Schrödinger equation (NLS) with a point-concentrated nonlinearity is a model Hamiltonian dispersive system in which the free one-dimensional Schrödinger equation is coupled, at a single point, to a nonlinear oscillator:

$$i\dot\psi(x,t)=-\psi''(x,t)-\delta(x)F(\psi(0,t)),\qquad x\in\mathbb R,$$

where $\delta(x)$ is the Dirac delta and the oscillator force $F$ derives from a potential, $F(\psi)=-\nabla U(\psi)$. When $U$ is invariant under phase rotations (the $U(1)$ symmetry), i.e. $F(\psi)=a(|\psi|^2)\psi$ with real $a\in C^1$, the equation conserves the charge $\int|\psi|^2\,dx$ and the energy $\frac12\int|\psi'|^2\,dx+U(\psi(0))$, and admits solitary waves (standing bound states) $\psi(x,t)=\psi_\omega(x)e^{i\omega t}$, which here take the explicit form $\psi_\omega(x)=Ce^{i\theta}e^{-\sqrt{\omega}|x|}$ with constant phase $\theta\in[0,2\pi]$ and, for $C>0$, amplitude and frequency related by $a(C^2)=2\sqrt{\omega}$; the family parametrized by $(C,\theta)$ (equivalently $(\omega,\theta)$) forms a two-dimensional solitary manifold in the finite-energy phase space. The central question of soliton dynamics is whether a solution converges, as $t\to\pm\infty$, to a single solitary wave plus dispersing radiation — a sum of another solitary wave and a solution $W(t)\Phi_\pm$ of the free Schrödinger equation, where $W(t)$ denotes the free Schrödinger group. Such asymptotic stability was first established by Soffer and Weinstein for weakly nonlinear multichannel scattering models and by Buslaev and Perelman for one-dimensional NLS with data near a stable soliton.

Buslaev, Komech, Kopylova and Stuart (2008) proved this for the point-coupled model: for data sufficiently close to the stable part of the solitary manifold, and under the spectral condition that the linearized operator has no discrete spectrum except the zero eigenvalue forced by the phase symmetry, the solution satisfies

$$\psi(\cdot,t)=\psi_{\omega_\pm}e^{i\omega_\pm t}+W(t)\Phi_\pm+r_\pm(t),\qquad \|r_\pm(t)\|_{C_b(\mathbb R)\cap L^2(\mathbb R)}=O(|t|^{-1/2}),$$

where $C_b(\mathbb R)$ is the space of bounded continuous functions. The point model is simple enough that all spectral properties of the linearization can be computed exactly. Writing $\nu=a'(C^2)C^2/a(C^2)$ for the normalized derivative of the nonlinearity at the solitary-wave amplitude $C$, the linearly stable range is $\nu<1$: for $\nu>1$ the wave is linearly unstable; for $\nu\in(-\infty,0)\cup(0,1/\sqrt2)$ the linearization has no additional discrete spectrum (the regime of the 2008 theorem); and for $\nu\in(1/\sqrt2,1)$ it acquires a pair of purely imaginary eigenvalues $\pm i\mu$ with $\mu<\omega$ — an oscillatory internal mode, i.e. a discrete non-dispersing oscillatory component of the linearized dynamics. The paper concludes by posing two open problems: whether the above asymptotics hold for any finite-energy solution (global attraction to the solitary manifold), which the authors hoped to attack with techniques then being developed for the Klein–Gordon equation with the same point nonlinearity; and the extension to the internal-mode regime. The subsequent history is asymmetric. The internal-mode regime was largely settled for the resonant part of the range by Komech, Kopylova and Stuart (2012) via a nonlinear Fermi golden rule condition — a non-degeneracy of the coupling between the second harmonic of the internal mode and the continuous spectrum, where the golden rule is the standard mechanism by which a discrete mode is damped through resonant energy leakage into radiation. Global attraction, in contrast, was proved in the Komech program for the Klein–Gordon, wave, and Dirac analogs with concentrated nonlinearities but has resisted all attempts in the Schrödinger case, whose one-dimensional radiation decays only like $t^{-1/2}$, too slowly for the local-energy-decay and tauberian arguments (methods of harmonic analysis that convert averaged decay of the solution's Fourier transform into pointwise decay in time) that drive the relativistic proofs.

Problem Statement

The overarching objective is to complete the soliton-plus-radiation asymptotics of the one-dimensional Schrödinger equation coupled to a nonlinear oscillator,

$$i\dot\psi(x,t)=-\psi''(x,t)-\delta(x)\,a(|\psi(0,t)|^2)\,\psi(0,t),\qquad x\in\mathbb R,$$

with a $U(1)$-invariant $C^2$ potential, i.e. $F(\psi)=a(|\psi|^2)\psi$ with real $a\in C^1$ and $U(z)\ge A-B|z|^2$ for some constants $A\in\mathbb R$ and $B>0$, in the two directions left open by Buslaev–Komech–Kopylova–Stuart (2008).

(1) Global attraction (principal part). Prove, or disprove, that for every admissible nonlinearity $a$, every finite-energy solution $\psi\in C(\mathbb R,H^1(\mathbb R))$ admits the long-time asymptotics

$$\psi(\cdot,t)=\psi_{\omega_\pm}e^{i\omega_\pm t}+W(t)\Phi_\pm+r_\pm(t),\qquad t\to\pm\infty,$$

where $\psi_{\omega_\pm}(x)=C_\pm e^{i\theta_\pm}e^{-\sqrt{\omega_\pm}|x|}$, $C_\pm\ge0$, $\theta_\pm\in[0,2\pi]$, is the solitary wave of frequency $\omega_\pm>0$ — the constant phase $\theta_\pm$ being part of the solitary wave by the $U(1)$ symmetry, and, for $C_\pm>0$, amplitude and frequency being related by $a(C_\pm^2)=2\sqrt{\omega_\pm}$ (the two-parameter family of these waves is the solitary manifold; following the source's convention, $C=0$ gives the zero function, which is a solitary wave for every admissible nonlinearity since $F(0)=0$) — $W(t)$ denotes the free one-dimensional Schrödinger group, the asymptotic scattering states are $\Phi_\pm\in C_b(\mathbb R)\cap L^2(\mathbb R)$, and the remainder $r_\pm(t)\to0$ in the global norm $C_b(\mathbb R)\cap L^2(\mathbb R)$ (the rate $O(|t|^{-1/2})$ of the local theorem is expected, but the essential claim is the decomposition with vanishing remainder), under no restriction on the initial data beyond finite energy — no closeness to the solitary manifold and no smallness assumption. Under this convention, the zero solution $\psi\equiv0$, and the degenerate linear case $a\equiv0$ (in which every solution is a free solution), satisfy the decomposition trivially with the zero solitary wave and hence do not constitute counterexamples; a disproof requires exhibiting an admissible nonlinearity and a finite-energy solution that admits no such decomposition for any $C\ge0$, $\theta_\pm\in[0,2\pi]$, $\omega_\pm>0$, and $\Phi_\pm\in C_b(\mathbb R)\cap L^2(\mathbb R)$.

(2) The non-resonant internal-mode regime (complementary part). Prove, or disprove, that the local asymptotic-stability theorem (data in a fixed neighborhood of a linearly stable solitary wave) extends to the remaining parameter window $\nu\in(\sqrt2(1+\sqrt3)/4,\,1)$, where $\nu=a'(C^2)C^2/a(C^2)$ is the normalized derivative of the nonlinearity at the solitary-wave amplitude $C$: in this window the linearization possesses an oscillatory internal mode — a pair of purely imaginary eigenvalues $\pm i\mu$ — whose second harmonic $2\mu$ lies below the continuous-spectrum edge $\omega$, so that the Fermi-golden-rule mechanism of Komech–Kopylova–Stuart (2012), the damping of the internal mode by resonant energy leakage into radiation, is unavailable; a disproof requires an admissible nonlinearity in this window together with data in a fixed neighborhood of a linearly stable solitary wave for which the local asymptotics fail; the source's announced goal was coverage of the entire linearly stable range $\nu<1$.

Part (1) is the principal open question; part (2) is complementary to it and can be addressed independently. Part (2) extends the parameter coverage of the local theory to the remaining non-resonant open window — the threshold values $\nu=0$, $\nu=1/\sqrt2$, and $\nu=\sqrt2(1+\sqrt3)/4$, at which the corresponding spectral conditions degenerate to equalities, being borderline cases. The two parts are logically linked: an affirmative proof of part (1) supplies the qualitative decomposition but not the rate refinement that part (2) asserts, whereas a non-resonant-window counterexample — an admissible nonlinearity in the window together with data near a linearly stable solitary wave for which the local asymptotics fail — refutes both parts, its data lying within part (1)'s scope. The two parts share one scientific objective: the complete soliton-radiation description of all finite-energy dynamics of the model over its full linearly stable parameter range.

References

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