Three thresholds and a computer-assisted phase diagram for the spectral stability of periodic hydroelastic Whitham waves

Lingsen Meng 

We study the spectral stability of 2π-periodic travelling waves of the hydroelastic Whitham equation, a scalar model with the full linear dispersion relation of water waves under a thin elastic sheet (e.g. a floating ice cover), as a function of the flexural rigidity T. Besides the modulational cluster at the origin, the zero-amplitude spectrum has, on an interval of T, a collision of opposite Krein signature between the carrier and the half-wavenumber mode, and three thresholds with closed forms in tanh(1/2), tanh(1) and tanh(2) organise the picture: waves are modulationally unstable below T0 ≈ 0.16638 (the long-wave/short-wave resonance), unstable through this sideband collision on (T0, T1], T1 ≈ 0.23108, and spectrally stable above T1, so that the stability threshold is T1 and not the modulational threshold T0. We prove this phase diagram for all Bloch parameters, with explicit amplitude windows, by combining analytic reductions with remainder control and a computer-assisted proof in interval arithmetic at finite amplitude. For every amplitude 0 < a ≤ 0.01 (height H ≲ 0.02 of the depth) the waves are unstable for T in [0.0675, 0.16516] ∪ [0.1681, T1] and spectrally stable for T ≥ 0.23108 with an amplitude window that closes linearly at T1, and a rescaling by T^(-1/2) extends the stable statement uniformly to the purely flexural limit. Numerically, our computations reproduce those of Flamarion and Adriazola (2026) and suggest that the instabilities they report for 0.1 < T ≤ 0.7 at larger heights are the finite-amplitude continuation of the sideband collision rather than a modulational instability.