S. Dyachenko and D.E. Pelinovsky
Bifurcations of unstable eigenvalues for Stokes waves derived from conserved energy
Abstract:
We address Euler’s equations for irrotational gravity waves in an infinitely deep
fluid rewritten in conformal variables. Stokes waves are traveling waves with the smooth
periodic profile. In agreement with the previous numerical results, we give a rigorous proof
that the zero eigenvalue bifurcation in the linearized equations of motion for co-periodic
perturbations occurs at each extremal point of the energy function versus the steepness
parameter, provided that the wave speed is not extremal at the same steepness. We derive
the normal form for the unstable eigenvalues and, assisted with numerical approximation of
its coefficients, we show that the new unstable eigenvalues emerge only in the direction of
increasing steepness.
Keywords:
Euler's equations; conformal variables; Stokes waves; spectral stability; instability bifurcation; normal form;