Stokes waves at the critical depth are modulational unstable

TL;DR

This paper rigorously demonstrates that small amplitude Stokes waves are linearly unstable at the critical Whitham-Benjamin depth and nearby, resolving a long-standing open question about their stability in shallow water.

Contribution

It provides a rigorous mathematical proof of the instability of Stokes waves at the critical depth, including detailed eigenvalue behavior near the transition point.

Findings

Stokes waves at critical depth are linearly unstable.

Eigenvalues form a closed '8' shape in the transient regime.

Instability persists for depths slightly smaller than the critical depth.

Abstract

This paper fully answers a long standing open question concerning the stability/instability of pure gravity periodic traveling water waves -- called Stokes waves -- at the critical Whitham-Benjamin depth $ \mathtt{h}_{\scriptscriptstyle WB} = 1.363... $ and nearby values. We prove that Stokes waves of small amplitude $ \mathcal{O}( \epsilon ) $ are, at the critical depth $ \mathtt{h}_{\scriptscriptstyle WB} $, linearly unstable under long wave perturbations. This is also true for slightly smaller values of the depth $ \mathtt{h} > \mathtt{h}_{\scriptscriptstyle WB} - c \epsilon^2 $, $ c > 0 $, depending on the amplitude of the wave. This problem was not rigorously solved in previous literature because the expansions degenerate at the critical depth. In order to resolve this degenerate case, and describe in a mathematically exhaustive way how the eigenvalues change their stable-to-unstable nature along this shallow-to-deep water transient, we Taylor expand the computations of arXiv:2204.00809v2 at a higher degree of accuracy, derived by the fourth order expansion of the Stokes waves. We prove that also in this transient regime a pair of unstable eigenvalues depict a closed figure "8", of smaller size than for $ \mathtt{h} > \mathtt{h}_{\scriptscriptstyle WB} $, as the Floquet exponent varies.