Asymptotic stability of one-dimensional nonlinear Schrödinger solitary waves beyond the spectral assumptions: internal modes and edge resonances

Collot and Germain (2023; published in the Journal of the European Mathematical Society, 2025) proved global asymptotic stability of solitary waves of the one-dimensional focusing nonlinear Schrödinger equation for a general class of nonlinearities, under the spectral assumptions that the linearized operator around the wave has no internal modes (no nonzero eigenvalues) and no edge, i.e. threshold, resonance, for perturbations small in $H^1\cap L^{2,1}$. The problem they pose in their Perspectives section — and which Germain's 2024 review identifies as one of the two main gaps in the field — is to extend this theorem beyond these spectral hypotheses. Internal modes and edge resonances produce singular structures in the distorted Fourier representation of the perturbation, and their dynamical consequences (radiation damping versus secular energy exchange, slow decay or drift of the modulation parameters, breakdown of the modified-scattering ansatz) are understood only in special situations: the integrable cubic equation, even perturbations of the cubic soliton, small near-cubic solitary waves under a Fermi golden rule, pure powers with numerically supported spectral data, and nonlinearities vanishing to high order at the origin. Whether full asymptotic stability holds for general, non-integrable nonlinearities that are cubic at the origin when internal modes or edge resonances are present is open.

Background

The equation and its solitary waves. Consider the one-dimensional focusing nonlinear Schrödinger equation

$$i\partial_t v-\partial_x^2 v-F'(|v|^2)v=0,\qquad (t,x)\in\mathbb{R}\times\mathbb{R},$$

where $F:\mathbb{R}\to\mathbb{R}$ is smooth and has a non-degenerate local minimum at the origin. By the classical theory of Berestycki and Lions there is a branch $(\Phi_\omega)_{\omega\in(0,\omega_*)}$ of ground-state solitary waves: each $\Phi_\omega$ is positive, even, exponentially decaying, and unique up to translation. The equation is invariant under phase rotation, space translation, and Galilean boosts, which generates a four-parameter family of traveling solitary waves $e^{i(px+(p^2-\omega)t+\gamma)}\Phi_\omega(x+2pt-y)$. If $F''(0)\neq0$ the nonlinearity is cubic to leading order near the origin, and the equation is a perturbation of the cubic (focusing) NLS, which is completely integrable; if $F$ vanishes to higher order the nonlinearity is weaker at the origin.

Stability notions. Orbital stability — solutions that start near the soliton family remain near it up to the symmetries — is characterized by the Grillakis–Shatah–Strauss theory: it holds exactly when $c_\omega=\frac{d}{d\omega}\int\Phi_\omega^2\,dx>0$. Asymptotic stability is stronger: the solution must converge, up to modulation of the parameters, to a solitary wave plus dispersing radiation. Full asymptotic stability means convergence in $L^2(\mathbb{R})$ together with a scattering description of the radiation; in one dimension a cubic nonlinearity is long range, so the radiation does not scatter linearly but undergoes modified scattering, acquiring a logarithmic phase correction. Local asymptotic stability, proved by virial (energy-monotonicity) methods, asserts convergence only on arbitrarily large compact sets and cannot exclude small distant solitary waves.

The linearized operator. Writing the solution as a modulated soliton plus a perturbation and separating real and imaginary parts linearizes the equation to $\partial_t W=\mathcal{H}_\omega W$, where $\mathcal{H}_\omega$ is a $2\times2$ non-selfadjoint matrix Schrödinger operator. Its spectral structure, analyzed rigorously by Krieger and Schlag in the supercritical case and numerically by Chang, Gustafson, Nakanishi, and Tsai in general, is the decisive input: the essential spectrum is $(-\infty,-\omega]\cup[\omega,\infty)$; the kernel is spanned by the four tangent directions to the soliton family; and three kinds of degeneracies may occur. Internal modes are nonzero real discrete eigenvalues in $(-\omega,\omega)$; on the linearized flow they oscillate forever, so asymptotic stability, if true, must come from a nonlinear damping mechanism. Unstable modes are eigenvalues with nonzero imaginary part, present for supercritical powers; they force finite-codimension center-stable manifolds (Krieger–Schlag) and exponential growth off the manifold, but the source paper deems their treatment by stable/unstable manifolds within reach. An edge resonance (or threshold resonance) is a bounded, non-square-integrable generalized eigenfunction of $\mathcal{H}_\omega$ at the spectral edge $\pm\omega$; it degrades the local decay of the linearized group, which is the backbone of the dispersive analysis. For pure powers $F(z)=z^{\sigma+1}$ the numerically computed picture is: cubic ($\sigma=1$) has an edge resonance; $1<\sigma<2$ have internal modes; $\sigma=2$ (the $L^2$-critical quintic equation, for which $c_\omega=0$) degenerates the kernel; and $\sigma>2$ have unstable modes.

The spectral method. The proof framework of the source paper combines: the distorted Fourier transform, a generalization of the Fourier transform built from the scattering theory of $\mathcal{H}_\omega$ that diagonalizes the linearized evolution; modulation, i.e. letting the soliton parameters evolve with the solution; and space-time resonance analysis of the nonlinear interactions in distorted Fourier variables. For a nonlinearity with an internal mode, the Fermi golden rule — the nonlinear transfer of energy from the discrete mode to the continuous spectrum — is the mechanism of radiation damping by which the mode amplitude may nevertheless decay.

Prior results. Buslaev and Perelman proved the first full asymptotic stability result for nonlinearities vanishing to order five at the origin, assuming no resonance and no internal modes; Buslaev and Sulem handled one internal mode under the same high-order hypothesis via radiation damping. Krieger and Schlag constructed stable manifolds for all supercritical powers. Cuccagna and Pelinovsky proved full asymptotic stability of the cubic soliton using complete integrability (inverse scattering), extended to $N$-soliton configurations by Saalmann and by Borghese, Jenkins, and McLaughlin. Martel introduced the virial approach for the cubic-quintic equation. Li and Lührmann proved full asymptotic stability of the cubic soliton under even perturbations by PDE methods, uncovering null-structure cancellations that tame the edge resonance. Rialland treated near-cubic equations for small solitary waves, without and with an internal mode, under a Fermi golden rule hypothesis. Cuccagna and Maeda proved related results for pure powers in windows around the cubic exponent, in part assuming the numerically computed spectral picture. Collot and Germain combined, for the first time, the distorted Fourier transform with modulation and renormalization, proving: if $c_\omega>0$, $\mathcal{H}_\omega$ has no internal modes and no edge resonance, then for data $\Phi_{\omega_0}+u_0$ with $\|u_0\|_{H^1}+\|u_0\|_{L^{2,1}}$ small (where $L^{2,1}$ is the $L^2$ space weighted by $\langle x\rangle$), the solution is global, asymptotically stable, and the radiation undergoes modified scattering if $F$ is cubic at the origin, standard scattering otherwise. The theorem gives no information when internal modes or edge resonances are present.

Problem Statement

Consider the one-dimensional focusing nonlinear Schrödinger equation

$$i\partial_t v-\partial_x^2 v-F'(|v|^2)v=0$$

on the real line, where $F$ is smooth with a non-degenerate local minimum at the origin, $F$ is allowed to be cubic at the origin ($F''(0)\neq0$), and the ground-state branch $(\Phi_\omega)_{\omega\in(0,\omega_*)}$ satisfies the orbital-stability condition $c_\omega=\frac{d}{d\omega}\int\Phi_\omega^2\,dx>0$. Let $\mathcal{H}_\omega$ denote the linearized operator around $\Phi_\omega$.

Collot and Germain proved full asymptotic stability — convergence of solutions starting near $\Phi_{\omega_0}$ in $H^1\cap L^{2,1}$ to a modulated solitary wave, with the radiation undergoing modified scattering (cubic case) or standard scattering (higher-order case) — under the spectral assumptions that $0$ is the only eigenvalue of $\mathcal{H}_\omega$ (in particular, no internal modes, i.e. no eigenvalues in $(-\omega,\omega)\setminus\{0\}$) and no edge resonance (no nonzero bounded solution of $\mathcal{H}_\omega f=\omega f$).

The problem is to remove these spectral hypotheses. Establish full asymptotic stability — or rigorously refute it — when $\mathcal{H}_\omega$ is allowed to have internal modes and/or an edge resonance at $\pm\omega$, for a class of nonlinearities that: is not the integrable cubic case; is not restricted to even initial data; is not restricted to the small-amplitude (small-frequency) soliton limit; and includes at least one nonlinearity that is cubic at the origin. The answer must hold on a nontrivial range of frequencies $\omega$ and must specify the asymptotic behavior of the radiation and of the modulation parameters (frequency, phase, boost, translation), in particular whether the nonlinear coupling to the continuous spectrum (the Fermi golden rule / radiation damping mechanism) forces the internal-mode amplitude to decay and whether the modulation parameters converge. A refutation must rigorously construct, within the stated class, solutions arbitrarily close to a solitary wave whose perturbation fails to converge to radiation plus a modulated soliton, or prove an alternative asymptotic description. The problem covers both degeneracies — internal modes and edge resonances — in the nonlinearities where each occurs. The $L^2$-critical quintic case ($c_\omega=0$), multi-soliton configurations, and the verification of spectral assumptions for concrete nonlinearities, which the source paper identifies as separate open problems, are not part of this problem's scope.

References

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