Long-Time Dynamics of Fractional Dispersive Equations

Definitions and mathematical statement

The supplied record concerns the fractional dispersive equation \[ \partial_t u(t,x)=\partial_x\left(|\partial_x|^\alpha u(t,x)-u(t,x)^p\right),\qquad x\in{\mathbb{R}}. \] Here $t$ denotes time, $x$ denotes the spatial variable, and $\alpha$ and $p$ are parameters whose admissible ranges are not supplied. The operator $|\partial_x|^\alpha$ is interpreted as the spatial Fourier multiplier \[ \widehat{|\partial_x|^\alpha f}(\xi)=|\xi|^\alpha\widehat f(\xi), \] whenever this expression defines an operator on the chosen function space. Its domain, including any necessary interpretation at $\xi=0$, must be specified after choosing $\alpha$ and the function space. The source does not specify whether $u$ is real or complex valued, or how $u^p$ is defined for noninteger $p$.

Research problem as supplied. Determine how the balance between dispersion, represented by $\partial_x|\partial_x|^\alpha u$, and nonlinearity, represented by $-\partial_x(u^p)$, depends on $\alpha$ and $p$ and governs long-time dynamics. The requested topics include global behavior, singularity formation, and the stability of solitary waves.

A traveling-wave profile $Q$ with speed $c\in{\mathbb{R}}$ gives a solution of the form $u(t,x)=Q(x-ct)$ when substitution satisfies the equation in the chosen solution sense. Which such profiles qualify as solitary waves requires additional localization or limiting conditions that are absent from the record. Stability likewise requires a specified topology, perturbation class, and notion of stability; the record does not distinguish orbital, asymptotic, or another form of stability.

The source supplies no initial-data class, regularity assumptions, solution concept, spatial decay conditions, time direction, or criterion for singularity formation. Consequently, it does not specify a universal or existential assertion over initial data, a parameter classification, or a particular stability conjecture. This is a statement of a research program with unresolved qualifications, rather than a uniquely quantified conjecture. These missing choices must be resolved before a precise theorem-like formulation can be made.

Short English statement

For a fractional dispersive equation on ${\mathbb{R}}$, investigate how $\alpha$ and $p$ influence long-time behavior, singularity formation, and solitary-wave stability. The supplied formulation leaves parameter ranges, admissible data, solution spaces, and stability criteria unspecified, so it identifies a research program rather than a single precise conjecture.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004188$, title: Long-Time Dynamics of Fractional Dispersive Equations; exact source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is 2019.
• Supplied report citation: Mathematical Theory of Water Waves (2020), Oberwolfach Reports, pp. 1919--1979, DOI 10.4171/owr/2019/32. Supplied source URL: https://doi.org/10.4171/owr/2019/32. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No page or passage specific to this formulation is supplied.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/owr/2019/32.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem open and describes it as a motivational formulation of an unresolved long-time-dynamics problem for a dispersive nonlinear PDE, rather than a theorem or uniquely specified conjecture. This is supplied dated curation, not an independently verified literature assessment. The supplied background requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and states that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. These are supplied editorial qualifications, not independent mathematical verification of this draft.