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Gray solitons are a one-parameter family of solutions to the one-dimensionalnon-linear Schr\"odinger equation (NLSE) with positive cubic nonlinearity, asfound in repulsively interacting dilute Bose-Einstein condensates orelectromagnetic waves in the visible spectrum in waveguides described byGross-Pitaevskii mean field theory. In two dimensions these solutions to theNLSE appear as a line or plane of depressed condensate density or lightintensity, but numerical solutions show that this line is dynamically unstableto `snaking': the initially straight line of density or intensity minimumundulates with exponentially growing amplitude. To assist future studies ofquantum mechanical instability beyond mean field theory, we here pursue anapproximate analytical description of the snake instability withinBogoliubov-de Gennes perturbation theory. Within this linear approximation thetwo-dimensional result applies trivially to three dimensions as well,describing buckling modes of the low-density plane. We extend the analyticalresults of Kuznetsov and Turitsyn [Sov. Phys. JETP \textbf{67}, 1583 (1988)] toshorter wavelengths of the `snake' modulation and show to what extent the snakemode can be described accurately as a parametric instability, in which theposition and grayness parameter of the initial soliton simply become dependenton the transverse dimension(s). We find that the parametric picture remainsaccurate up to second order in the snaking wave number, if the snaking solitonis also dressed by an outward-propagating sound wave, but that beyond secondorder in the snaking wave number the parametric description breaks down.

Bogoliubov-de Gennes theory of the "snake" instability of gray solitons in higher dimensions