Higher-Dimensional Frenkel–Kontorova Traveling Waves

Definitions and mathematical statement

A traveling heteroclinic wave is a solution represented by a profile translating at a constant speed and approaching distinct equilibrium states at its two ends. In scalar profile notation, the heteroclinic boundary conditions are \[ \lim_{\xi\to-\infty}U(\xi)=a_-,\qquad \lim_{\xi\to+\infty}U(\xi)=a_+,\qquad a_-\ne a_+, \] where $U\colon{\mathbb{R}}\to{\mathbb{R}}$ is the profile and $a_-,a_+$ are equilibrium states associated with the two wells. This notation explains the terminology; the supplied record does not specify the governing equation or whether this scalar profile description is the intended higher-dimensional formulation.

Extension problem. Can the one-dimensional Frenkel--Kontorova existence theorem for traveling heteroclinic waves in suitable smooth anharmonic two-well potentials be extended to higher spatial dimensions $d\ge 2$?

The word "suitable" retains an unresolved qualification from the source. The supplied record does not give the precise one-dimensional theorem, its potential hypotheses, its admissible wave speeds, or the higher-dimensional equations and wave geometry. Consequently, a fully quantified existence statement cannot be extracted from this record alone. In particular, no assumptions about equal well depths, nondegenerate minima, propagation direction, or nonzero speed are supplied here.

Short English statement

The problem asks for a higher-dimensional analogue of a one-dimensional Frenkel--Kontorova theorem producing traveling waves that connect two distinct equilibrium states in a suitable smooth anharmonic two-well potential. The equations, potential assumptions, speed restrictions, and intended higher-dimensional wave geometry remain unspecified.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30004185$, titled "Higher-Dimensional Frenkel--Kontorova Traveling Waves," in the Oberwolfach Reports Open Problems collection. Exact source problem code: catalogue entry. The record gives proposed year 2019 and names no proposer.
• Supplied source citation: Mathematical Theory of Water Waves (2020), Oberwolfach Reports, pp. 1919--1979, DOI 10.4171/owr/2019/32. Original report: https://doi.org/10.4171/owr/2019/32. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Buffoni--Schwetlick--Zimmer, "heteroclinic waves" (the supplied reference label), https://doi.org/10.1016/j.matpur.2019.01.002.
• Supplied literature assessment dated exactly 2026-08-22: the higher-dimensional extension is posed as a question, and the cited primary work establishes only the one-dimensional-chain result. The record labels the problem open. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; the status and cited claims require independent verification.