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

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