Asymptotic Stability of Orbitally Stable NLS Ground States

Definitions and mathematical statement

Consider \[iu_t+\Delta u+\beta(|u|^2)u=0\qquad (t,x)\in{\mathbb{R}}\times{\mathbb{R}}^3,\] with standing waves $u(t,x)=e^{it\omega}\varphi_\omega(x)$, $\omega>0$, satisfying $-\Delta\varphi_\omega+\omega\varphi_\omega-\beta(|\varphi_\omega|^2)\varphi_\omega=0$. The nonlinearity and its regularity/growth hypotheses are not specified. In the usual variational setting, if $B(s)=\int_0^s\beta(a)\,da$, the energy and action are $E(v)=\tfrac12\int_{{\mathbb{R}}^3}(|\nabla v|^2-B(|v|^2))$ and $S_\omega(v)=E(v)+\tfrac\omega2\|v\|_2^2$; a ground state is a least-action nonzero stationary solution. This terminology presupposes that these objects are defined for the model. Orbital stability means that sufficiently small $H^1$ perturbations remain close to the appropriate symmetry orbit for all forward times; the allowed orbit depends on the symmetry class. The conjectural implication in the source is that every orbitally stable ground state is asymptotically stable. Its displayed asymptotic assertion asks whether every sufficiently nearby finite-energy solution admits \[\omega_0>0,\qquad\theta\in C^1({\mathbb{R}}_+;{\mathbb{R}}),\qquad h_0\in H^1({\mathbb{R}}^3)\] such that \[\left\|u(t)-e^{i\theta(t)}\varphi_{\omega_0}-e^{it\Delta}h_0\right\|_{H^1({\mathbb{R}}^3)}\longrightarrow0\qquad(t\to\infty).\] The source gives no additional limiting condition on $\theta'$; none is inserted. The limiting frequency and radiation may depend on the solution.

The original statement explicitly contains the qualification for symmetric solutions alongside this formula, whereas the clean/current version omits it. That qualification is essential to preserve: the displayed limiting solitary wave has no translation, velocity, or moving-center parameter. For unrestricted nearby data, translation and Galilean symmetries require attention; the fixed-center formula cannot silently be used as a full nonsymmetric modulation statement. The source does not define symmetric or say precisely which part of the conclusion the parenthetical qualifies, so neither a radial restriction nor a translated/boosted replacement is invented. The supplied 2026-08-21 triage labels the record partially solved, reports asymptotic stability for symmetric finite-energy perturbations under the Fermi Golden Rule and related spectral hypotheses, and describes the unrestricted conjecture as conditional and model-dependent.

Short English statement

Does orbital stability of an NLS ground state force convergence to a solitary wave plus free radiation? The original qualifies the displayed fixed-center formula by symmetry, a restriction missing from the cleaned version; nonlinearity and spectral hypotheses are also unstated.

Sources

• Local UnsolvedMath registry, permanent ID 30000924, source problem catalogue entry, Asymptotic Stability of Orbitally Stable NLS Ground States, proposed year 2008; no individual proposer supplied.
• Mini-Workshop: Attraction to Solitary Waves and Related Aspects of Physics (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/08, pp. 367--400: https://doi.org/10.4171/owr/2008/08. Supplied literature label: original OWR report.
• Supplied literature label: Cuccagna--Mizumachi theorem, https://arxiv.org/abs/0711.4195.
• Supplied literature label: 2020 survey, https://arxiv.org/abs/2009.00573.