A. Aghayev, E. Burnes, and D. E. Pelinovsky

Stokes waves in a finite-depth fluid via a conformal transformation

Abstract:
Euler equations for a spatially periodic surface in a two-dimensional fluid of a finite depth are reduced to a system of evolution equations after a conformal transformation. Compared to the previously developed formal approach, we show that the system of evolution equations is complemented by two constraints which determine the new depth in conformal variables and the Bernoulli integration constant. A locally unique continuation of the traveling Stokes waves with respect to the wave speed is studied by using arguments based on the implicit function theorem. We show that the constraints on the traveling wave profile recover the correct small-amplitude expansion of Stokes waves in physical variables. We also prove for the co-periodic stability problem that the real unstable eigenvalues may bifurcate from the zero eigenvalue only at a critical point of the wave energy. Finally, we perform numerical approximations of the wave profiles for both small and large amplitudes and illustrate different bifurcations along the solution family.

Keywords:
Euler's equations; conformal variables; Stokes waves; existence; spectral stability; numerical approximations.