Solitary-Wave Stability for the Lugiato–Lefever Equation

Definitions and mathematical statement

Let $\psi$ denote a complex-valued field, let $i^2=-1$, and let $|\psi|$ denote its complex modulus. The supplied equation is the Lugiato--Lefever equation \[ \partial_t\psi=-i\beta\,\partial_x^2\psi-(1+i\alpha)\psi+i|\psi|^2\psi+F. \] Here $t$ and $x$ are the time and spatial variables, respectively, and $\alpha,\beta,F$ are parameters. The source record does not specify their admissible values or whether each parameter is real or complex; it also does not specify the spatial domain or boundary conditions.

Requested mathematical target. Determine the stability or instability of the solitary-wave solutions of this equation throughout the intended parameter and solution family.

This target remains underdetermined by the supplied record. In particular, it does not define a solitary wave, specify whether the waves are stationary or traveling, prescribe their asymptotic background or decay, or identify the perturbation class and notion of stability. Spectral, linear, nonlinear, orbital, and asymptotic stability are distinct possible targets, and none is selected here. A precise quantified stability assertion requires these choices, including the solution regularity, topology, time direction, and treatment of any symmetries. No particular choice of these missing hypotheses is imposed in this draft.

Short English statement

Determine which solitary waves of the displayed Lugiato--Lefever equation are stable or unstable. The supplied record leaves the parameter regime, spatial setting, solitary-wave family, and meaning of stability unspecified, so it does not determine a unique precise stability problem.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004763$, title: Solitary-Wave Stability for the Lugiato--Lefever Equation; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. The record gives proposed year $2021$ and supplies no named proposer.
• Supplied source citation: Dynamics of Waves and Patterns (2022), Oberwolfach Reports, pp. 1965--2026, DOI 10.4171/owr/2021/37. https://doi.org/10.4171/owr/2021/37. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The original extracted target is to determine the stability of solitary waves for the considered nonlinear wave equation; the supplied corrected statement identifies that equation as the Lugiato--Lefever equation displayed above.
• Supplied literature source labeled Lafortune--Pelinovsky: https://doi.org/10.1016/j.physd.2022.133477.
• Supplied literature source labeled Oberwolfach report: https://ems.press/content/serial-article-files/46915.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that existence and criteria for several periodic and solitary waves are known, while stability of solitary waves in the general family described by the problem remains largely unresolved. This is supplied dated curation derived from the OWR material, not an independently verified assessment; the record explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked directly during its curation. This is a supplied provenance claim, not a claim that the present draft independently consulted or verified that report.