Asymmetric generalized Agnesi branch and homoclinic correspondence

Determine whether asymmetric solutions exist for generalized Agnesi forcing with n=4 and, if they do, establish whether their branch intertwines with the symmetric branch and characterize the relationship between the forcing profile and the number of homoclinics that can be emitted.

Background

The authors were unable to compute asymmetric solutions for the generalized Agnesi topography. They conjecture that any such branch would intertwine with the computed symmetric branch, in a structure reminiscent of homoclinic snaking, and identify the relation between forcing decay and the number of emitted homoclinics as an additional unresolved issue.

References

We were unable to compute asymmetric solutions for this topography, and the question of their existence is left to future work. We make the tentative conjecture that if a branch of asymmetric solutions does exist, then it would intertwine with the branch in figure~\ref{fig:fr_1_01_agnesi_n_4_kdv}. The winding of the branch is reminiscent of the well-known phenomenon of homoclinic snaking (e.g. ). This issue and understanding the correspondence between the forcing and number of homoclinics allowed to be emitted is left to a future study.

— 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”