Asymmetric Periodic Water Waves in Finite Depth

RESUMO

We consider the full two-dimensional water-wave equations in finite depth, with and without surface tension. Water waves have been studied mathematically for more than two centuries. Reflection symmetry remains a common assumption in constructions of periodic steady waves, although asymmetric profiles have been observed in experiments and numerical computations. We construct families of periodic stationary solutions whose free surfaces have no axis of reflection, in both the pure-gravity and capillary--gravity cases. To the best of our knowledge, this provides the first rigorous construction of asymmetric periodic pure-gravity waves for the full two-dimensional water-wave problem in finite depth. The proof combines localized vorticity laws with a Lyapunov--Schmidt reduction. In the pure-gravity case, we introduce a small background shear flow to obtain an invertible boundary operator and couple the interior and free-surface approximations at leading order.