Long-Time Resonant Dynamics on Large Domains

Definitions and mathematical statement

Let $L>0$ denote the characteristic size of a spatial domain $\Omega_L$, with the large-domain regime understood as $L\to\infty$. Write $u_L$ for a solution of the nonlinear dispersive equation under consideration on $\Omega_L$. These symbols specify notation only: the supplied record does not fix the equation, spatial dimension, domain geometry, boundary conditions, initial data, or their dependence on $L$.

The Euclidean approximation refers here to approximating the evolution on $\Omega_L$ by a corresponding evolution on Euclidean space. The record supplies neither the correspondence between the two solutions nor a norm, error bound, or time interval defining validity or breakdown. Likewise, resonant time scales is the source's terminology; no resonance relation or time-scale formula is specified.

Research question. For nonlinear dispersive equations on domains of very large size $L$, describe the dynamics after the Euclidean approximation ceases to be valid, and determine whether an effective evolution can still describe those dynamics precisely on longer, resonant time scales.

A precise instance of this question requires specification of the equation, domain family, boundary conditions, admissible data, parameter regime, comparison maps, approximation topology, effective evolution, and time scales. The supplied record does not determine these choices or whether the intended assertion concerns every admissible model or particular classes. Consequently, it does not specify a single quantified theorem or conjecture. Its broader interest in general boundary conditions must be retained; periodic nonlinear Schrödinger equations are mentioned as a setting of supplied partial results, not as the exclusive scope of the question.

Short English statement

Determine whether effective equations describe nonlinear dispersive waves on large domains after approximation by Euclidean evolution breaks down, particularly over longer resonant times. The source leaves the model and approximation criteria unspecified. Its supplied assessment reports partial progress for periodic nonlinear Schrödinger equations and an unresolved broader question for general boundary conditions; this assessment requires independent verification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003149$, title Long-Time Resonant Dynamics on Large Domains, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied.
• Supplied report citation: Nonlinear Evolution Problems (2017), Oberwolfach Reports, pp. 1517--1576, DOI 10.4171/owr/2016/27. https://doi.org/10.4171/owr/2016/27. The record describes the report as documenting workshop discussions and open problems. Its proposed year and year tag are 2016, while the supplied report title uses 2017; this dating discrepancy is unresolved.
• Supplied literature reference: Germain--Hani, analysis of the continuous-resonant equation. https://doi.org/10.1093/imrn/rnx156.
• Supplied literature reference: large-box NLS effective dynamics. https://arxiv.org/abs/1610.03824.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem partially solved and states that positive results describe effective dynamics for periodic nonlinear Schrödinger equations on large boxes on specified resonant time scales, while the broader question for general boundary conditions remains unresolved. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that its mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. This is a supplied provenance claim, not independent mathematical verification of this draft or of the source formulation.