Asymptotic behavior of branch-merging Froude numbers

Determine whether the sequence of Froude numbers at which successive fKdV-trench branches merge diverges to infinity or converges to a finite limiting value.

Background

For the fKdV model with trench forcing, the authors observe that branches merge as the Froude number increases and hypothesize that pairs of successive branches merge at an increasing sequence of Froude numbers. The unresolved issue is the asymptotic behavior of that sequence.

References

Whether $Fr_{n}\to \infty$ as $n\to\infty$ or whether it approaches a finite value is an open question.

— Homoclinic-shedding in the steady forced water-wave problem  (2609.04888 - Keeler et al., 4 Sep 2026) in Section 6, Discussion

One possible reason for the failure is that the numerical domain is not wide enough to accommodate the emission of a further pair of solitary waves. We hypothesise that the branch will continue indefinitely, and hence for a finite range of $a$ there could be an infinite number of solutions.

— Homoclinic-shedding in the steady forced water-wave problem  (2609.04888 - Keeler et al., 4 Sep 2026) in Section 5.2, subsection “Generalised Agnesi forcing with n=4”