Symmetry-breaking bifurcation on the Gaussian-forcing Euler branch

Determine whether a symmetry-breaking bifurcation occurs at the L2 limit point on the B1 solution branch of the fully nonlinear steady Euler system for Gaussian topography.

Background

For Gaussian topography at Fr=1.01, the fully nonlinear Euler system exhibits solution branches B0, B1, and B2. The L2 limit point on B1 has a profile resembling the symmetry-breaking point identified in the fKdV trench model, but numerical difficulties prevented continuation and direct verification of the bifurcation.

References

While numerical challenges prevented us from further detailed investigation here, it is tempting to conjecture that there is a symmetry-breaking bifurcation at this point.

— Homoclinic-shedding in the steady forced water-wave problem  (2609.04888 - Keeler et al., 4 Sep 2026) in Section 5.1, subsection “Solution space for fixed Fr”

In addition, we would expect to see asymmetric solutions over smooth topological topographies, such as the Gaussian or Agnesi studied here; we leave this for future work.

— Homoclinic-shedding in the steady forced water-wave problem  (2609.04888 - Keeler et al., 4 Sep 2026) in Appendix, Section “Asymmetric solutions”

Further outstanding questions concern i) the temporal stability of various solution branches, a point of particular interest for dynamical simulations in the unsteady setting, and ii) what is the solution structure for different forcing types for subcritical flow, $Fr<1$.

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