Papers › Stokes waves at the critical depth are modulational unstable
Stokes waves at the critical depth are modulational unstable
Massimiliano Berti, Alberto Maspero, Paolo Ventura
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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 h_(WB) = 1.363... and nearby values. We prove that Stokes waves of small amplitude 𝒪( ϵ) are, at the critical depth h_(WB), linearly unstable under long wave perturbations. This is also true for slightly smaller values of the depth h > h_(WB) - c ϵ², $ 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 h > h_(WB), as the Floquet exponent varies.
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