For p=2,3,4, the coefficient beta_1^{(p)}(h) controlling the exponentially small width of the p-th Stokes-wave isolas has explicit deep-water asymptotics, namely (3*sqrt(3)/64)e^{-h/2}, (2*sqrt(2)/3)e^{-2h}, and -(5*sqrt(5)/(8*sqrt(3)))e^{-2h}.
First isola of modulational instability of Stokes waves in deep water
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We prove high-frequency modulational instability of small-amplitude Stokes waves in deep water under longitudinal perturbations, providing the first isola of unstable eigenvalues branching off from $\mathtt{i}\frac34$. Unlike the finite depth case this is a degenerate problem and the real part of the unstable eigenvalues has a much smaller size than in finite depth. By a symplectic version of Kato theory we reduce to search the eigenvalues of a $2\times 2$ Hamiltonian and reversible matrix which has eigenvalues with non-zero real part if and only if a certain analytic function is not identically zero. In deep water we prove that the Taylor coefficients up to order three of this function vanish, but not the fourth-order one.
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On higher order isolas of unstable Stokes waves
For p=2,3,4, the coefficient beta_1^{(p)}(h) controlling the exponentially small width of the p-th Stokes-wave isolas has explicit deep-water asymptotics, namely (3*sqrt(3)/64)e^{-h/2}, (2*sqrt(2)/3)e^{-2h}, and -(5*sqrt(5)/(8*sqrt(3)))e^{-2h}.