Gray solitons on the surface of water at kh = 1.363

Existence, formation and dynamics of surface gravity water waves, are studied, in the form of gray solitons, when the characteristic parameter $kh$ - where $k$ is the wavenumber and $h$ is the undistorted water's depth - takes the critical value $kh=1.363$. In this case, the nonlinearity coefficient of the pertinent nonlinear Schr\"odinger (NLS) equation vanishes, and the proper model becomes a quintic NLS with higher-order nonlinear corrections. It is shown that this model admits approximate gray soliton solutions obeying an effective Korteweg-de Vries equation and that two types of gray solitons exist: fast and slow, with the latter being almost stationary objects. Our analytical predictions are in excellent agreement with direct numerical simulations.