Dark solitons for an extended quintic nonlinear Schr\"odinger equation: Application to water waves at kh = 1.363

We study the existence, formation and dynamics of gray solitons for an extended quintic nonlinear Schr\"odinger (NLS) equation. The considered model finds applications to water waves, when the characteristic parameter $kh$ - where $k$ is the wavenumber and $h$ is the undistorted water's depth - takes the critical value $kh=1.363$. It is shown that this model admits approximate dark soliton solutions emerging from an effective Korteweg-de Vries equation and that two types of gray solitons exist: fast and slow, with the latter being almost stationary objects. Analytical results are corroborated by direct numerical simulations.