Langmuir wave collapse with anisotropic contraction rates

The anisotropic collapse of Langmuir waves is investigated within the Zakharov equations and the nonlinear Schr\"odinger equation descriptions. By means of a transformation group we exhibit self-similar anisotropic solutions characterized by two different contraction rates. We show the existence of two infinite sets of anisotropic collapsing solutions, corresponding to oblate- or oblong-shaped collapsing structures. The usual isotropic collapsing solution is found to be a natural limit of these two sets. The two classes of anisotropic solutions are furthermore shown to be bounded by two particular solutions whose validity extends from the subsonic to the supersonic regime; the spatial profiles of these so-called trans-sonic solutions are analytically and numerically derived.