Borderline Soliton–Potential Interactions

Definitions and mathematical statement

Let $m\in\{2,3,4\}$, $\varepsilon>0$, and $0\leq\lambda<1$. Let $a\in C^\infty({\mathbb{R}};(0,\infty))$ satisfy \[ \lim_{s\to-\infty}a(s)=1, \qquad \lim_{s\to+\infty}a(s)=2. \] Consider the equation \[ \partial_tu+\partial_x\!\left(\partial_x^2u-\lambda u+a(\varepsilon x)u^m\right)=0, \qquad (t,x)\in{\mathbb{R}}\times{\mathbb{R}}. \] The supplied record singles out a soliton-like solution with incoming asymptotic description \[ u(t,\cdot)\sim Q\left(\cdot-(1-\lambda)t\right) \qquad\text{as }t\to-\infty. \] Here $Q$ is the profile named in the record; neither its defining equation nor its normalization is supplied. The symbol $\sim$ records the stated asymptotic description only: the source record does not specify a norm, topology, or convergence rate.

The question is to determine the long-time asymptotic behavior of this solution as $t\to+\infty$ in the borderline case \[ \lambda=\widetilde\lambda, \] where $\widetilde\lambda$ is described as the threshold separating refraction from reflection. The record supplies no definition or characterization of $\widetilde\lambda$, no mathematical definitions of refraction and reflection, and no precise outgoing asymptotic criterion. It also leaves the solution class and scalar field unspecified. Consequently, this is a conditional statement of the supplied target, requiring these missing specifications before it becomes a fully precise asymptotic problem; existence and uniqueness of the designated solution are not asserted here.

Short English statement

For the stated variable-coefficient dispersive equation, determine the future behavior of the solution described as an incoming soliton when $\lambda$ equals the threshold between refraction and reflection. The profile, threshold, solution class, and meanings of the asymptotic descriptions require clarification.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001591$, title Borderline Soliton--Potential Interactions, exact source problem code catalogue entry, collection Oberwolfach Reports Open Problems. The proposed year is recorded as 2010; no proposer is named.
• Supplied report citation: Nonlinear Waves and Dispersive Equations (2011), Oberwolfach Reports, pp. 2393--2463, DOI 10.4171/owr/2010/41. Source URL and supplied literature-source link: https://doi.org/10.4171/owr/2010/41. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or contribution author is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied status is open; the assessment states that the extracted uniqueness question is under-specified and that no general resolution of the equal-parameter case was located. This is supplied dated curation, not independently verified. The record also labels the statement corrected_verified and reports that it was checked against the official report; that verification claim is supplied metadata, not an independent verification here. The supplied curation requests independent verification of time-sensitive status and cited claims.