Uniform Low-Regularity Schemes for Coupled Klein–Gordon Systems

Definitions and mathematical statement

The supplied research problem asks for an extension of uniformly accurate, low-regularity numerical approximations for the Klein--Gordon equation to more general Klein--Gordon-type systems, including coupled Klein--Gordon equations and Klein--Gordon--Schrödinger systems.

Uniform accuracy means that an approximation error bound remains valid with constants independent of the parameter or parameters over the intended parameter range. Schematically, if $\lambda\in\Lambda$ denotes such parameters, $\delta>0$ a discretization scale, and $E_\lambda(\delta)$ an error in a specified norm over a specified time interval, an algebraic uniform accuracy requirement would have the form \[ \sup_{\lambda\in\Lambda} E_\lambda(\delta)\leq C\delta^p, \qquad 0<\delta\leq\delta_0, \] where $p>0$, $C$, and $\delta_0$ are independent of $\lambda$. This is an explanatory template, not a bound specified by the source: the record supplies neither the parameters and their range nor the error norm, discretization, order, or admissible step restrictions. Low regularity refers to obtaining convergence estimates under weak regularity assumptions on the initial data and solutions; no function spaces or regularity thresholds are supplied here.

Statement of the research task. Extend the scalar Klein--Gordon approximation approach described by the source to more general Klein--Gordon-type systems, with coupled Klein--Gordon and Klein--Gordon--Schrödinger systems as examples, and establish approximation error estimates that combine low solution regularity with uniform accuracy in the relevant parameters.

Unresolved formulation qualifications. The supplied record does not state the scalar starting equation or approximation scheme, the target systems or coupling nonlinearities, the spatial domain and dimension, initial or boundary conditions, parameter scalings and ranges, solution class, error norm, convergence order, or time interval. It also does not specify whether temporal, spatial, or fully discrete approximations are intended. Consequently, it determines a research direction but does not determine a unique, fully quantified mathematical conjecture. The examples do not specify a universal class of systems to which the requested extension must apply.

Short English statement

Develop numerical approximations for coupled Klein--Gordon-type systems that converge for solutions of low regularity and retain their accuracy uniformly over the relevant parameter range. The supplied record leaves the equations, parameters, regularity assumptions, and convergence requirements unspecified. Its dated curation labels the problem partially solved and reports progress for Klein--Gordon--Schrödinger systems; these claims require independent review.

Sources

• Local supplied registry record: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30004762$, source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied. The recorded proposed year is $2021$.
• Supplied source citation: Dynamics of Waves and Patterns (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/37, pp. 1965--2026. https://doi.org/10.4171/owr/2021/37. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Cabrera Calvo--Schratz. https://doi.org/10.1016/j.cam.2022.114756. No article title or precise theorem reference is supplied.
• Supplied literature source: Oberwolfach report. https://ems.press/content/serial-article-files/46915.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially_solved and states that uniformly accurate low-regularity schemes have been extended to Klein--Gordon--Schrödinger systems, while the broader class of coupled Klein--Gordon-type systems is not covered in full. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified. Its verification note reports that the official Oberwolfach report was checked directly and that the clean statement preserves the complete target while restoring definitions and removing extraction artifacts or source-dependent wording. This is a supplied provenance claim; no source consultation or verification is asserted in this draft.