S. Dyachenko, R. Marangell, and D. E. Pelinovsky
Recurrent bifurcations of stability spectra for steep Stokes waves in a deep fluid
Abstract:
We study the modulational stability problem for the traveling periodic waves
(called Stokes waves) in an infinitely deep fluid by using pseudo-differential operators in
conformal variables. We derive the criteria and the normal forms for four bifurcations
which are repeated recurrently when the steepness of the Stokes wave is increased towards
the highest wave with the peaked profile. The four bifurcations are observed in the following
order: (a) new figure-8 bands appearing at each extremal point of speed, (b) degeneration
of figure-8 bands resulting in their vertical slope, (c) new circular bands around the origin
appearing at each period-doubling bifurcation, and (d) reconnection of figure-1 bands at
each extremal point of energy. Our work uses the analytic theory of Stokes waves developed previously
for Babenko’s equation. The novelty of our work is the analytic extension of the modulational stability problem
for singular pseudo-differential operators in terms of the Floquet
parameter. The derivation of the normal form uses some structural assumptions which are
known to be true for the Stokes waves. For the first and second bifurcation cycles, we
compute numerically with a higher-order accuracy the actual values of wave steepness for
which the structural assumptions are satisfied and the numerical coefficients of the normal
forms to show the excellent agreement between the normal form theory and the numerical
approximations of the spectral bands.
Keywords:
Euler's equations; conformal variables; Stokes waves; spectral stability; modulational stability;
instability bifurcations; normal forms;