Distinct high-frequency effects of crest and free-boundary degeneracies

Determine whether the free-boundary degeneracy and the crest degeneracy of the gradient-power compactons generate distinct high-frequency mechanisms in the linearised evolution.

Background

The linearised evolution generator is degenerate at the compacton free boundary for all compactons and, when r>1r>1, also at the crest because the derivative of the nonlinear flux vanishes there. Numerical Galerkin truncations show no detectable real part in the tested point spectra, but the paper emphasizes that this does not rule out high-frequency transient amplification or ill-posedness.

The authors explicitly identify as unresolved whether the two geometrically different degeneracies lead to different high-frequency behaviours. Resolving this would clarify well-posedness and stability beyond the self-adjoint spectral factor analyzed in the paper.

References

Equation~eq:E is particularly interesting in this respect because its generator degenerates at two geometrically distinct locations: at the free boundary for all compactons and, when $r>1$, at the crest where $\partial_zF_p(U)=0$. Determining whether these two degeneracies produce different high-frequency mechanisms is a natural continuation of the present work.

— Edge-cusped compactons from gradient-power nonlinear dispersion: exact transitions and singular spectral structure  (2610.00552 - Villatoro, 30 Sep 2026) in Section 7, Discussion, paragraph beginning “This self-adjoint information should not be confused with stability”