The last decade has seen remarkable progress on the numerical, asymptotic, and rigorous understanding of the instabilities of small-amplitude Stokes waves: surface water waves of permanent form, translating with constant velocity.

I will show the result of our numerical study on the instabilities of near-extreme amplitude Stokes waves, approaching the limiting wave with its internal corner of 120 degrees.

This is joint work with Sergey Dyachenko, Pavel Lushnikov, and Anastassiya Semenova.