Long-Time Action Preservation for Fourier Splitting of NLS

Definitions and mathematical statement

Let $K$ be a prime discretization size, let $\tau>0$ denote the time step, and let $\xi_j^n$ denote the numerical Fourier coefficients produced by the fully discrete Fourier splitting method for the one-dimensional cubic nonlinear Schrödinger equation referred to in the supplied record. Here $n\in{\mathbb{Z}}_{\geq 0}$ is the time-step index and $j$ ranges over the retained Fourier modes, whose index set is not specified. Define the numerical actions by \[ I_j^{K,n}=|\xi_j^n|^2. \] The supplied record states that an estimate \[ \left|I_j^{K,n}-I_j^{K,0}\right|\leq C\varepsilon \qquad\text{for }n\tau\leq T/\varepsilon \] is known in the underlying regime. The meaning of the positive parameter $\varepsilon$, the hypotheses of this estimate, and the permitted dependence of $C$ and $T$ are not supplied.

Long-time preservation question. Under the hypotheses intended for that estimate, does comparable preservation of the numerical actions persist to substantially longer times, such as times of order $\varepsilon^{-r}$ for large $r$? A same-order-error formulation of the requested extension would ask whether suitable positive constants $C_r,T_r$ give \[ \left|I_j^{K,n}-I_j^{K,0}\right|\leq C_r\varepsilon \qquad\text{whenever }n\tau\leq T_r\varepsilon^{-r}. \] This display describes a possible precise target, rather than a fully quantified conjecture supplied by the record. The record does not specify whether the extension is sought for every sufficiently large fixed $r$, for arbitrarily large $r$ under additional hypotheses, or for some particular large exponent. It also leaves unresolved the equation's spatial domain and boundary conditions, the precise splitting and Fourier discretization, the initial-data assumptions, restrictions on $\tau$ and $K$, and the required uniformity over modes and discretization parameters. These qualifications must be recovered before a complete mathematical statement can be fixed.

Short English statement

For a fully discrete Fourier splitting approximation of the one-dimensional cubic nonlinear Schrödinger equation with prime discretization size $K$, the supplied question asks whether numerical actions that remain within $C\varepsilon$ of their initial values until times of order $\varepsilon^{-1}$ enjoy comparable preservation until times of order $\varepsilon^{-r}$ for large $r$. The numerical scheme, parameter regime, and necessary quantifiers remain unspecified.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30001744$, source problem code catalogue entry, title: Long-Time Action Preservation for Fourier Splitting of NLS. The supplied record places this problem in the Oberwolfach Reports Open Problems collection and gives the proposed year as 2011; no individual proposer is supplied.
• Geometric Numerical Integration (2011), Oberwolfach Reports, pp. 825--900, DOI 10.4171/owr/2011/16. Supplied source URL and literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2011/16. The report is described as documenting mathematical workshops and their open-problem discussions; no exact problem page or internal locator is supplied.
• Supplied literature assessment, checked on 2026-08-21: the record labels the problem open, states that the report explicitly asks about preservation beyond the $O(\varepsilon^{-1})$ interval, and reports that no complete theorem for the stated regime was identified. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment.
• Supplied statement-verification metadata labels the statement corrected_verified and reports that the official report was checked to restore the equation, numerical actions, prime discretization size, known time scale, and requested longer scale. This records the supplied verification claim; it does not constitute independent verification or mathematical review of this draft.