Scalar wave collapse at critical dimension

The collapse of wave packets governed by the Zakharov equations is investigated at critical dimension. Their classical self-similar solutions are described by a linear time-dependent contraction scale ξ(t)=V(${\mathit{t}}_{\mathrm{*}}$-t) where ${\mathit{t}}_{\mathrm{*}}$ denotes the collapse time. We study two spherically symmetric versions of self-similar collapses, namely one corresponding to a vectorial electric field and another one relative to a scalar modelization of the Langmuir field. Each of these solutions can be regarded as a function of the collapse velocity V=-ξ̇. In the case of a vectorial electric field, the solutions are analytically shown to exist in the subsonic regime only, provided that the velocity V is lower than a critical velocity ${\mathit{V}}_{\mathrm{crit}}$, in agreement with previous numerical results. By contrast, the solutions of the scalar model are found to exist for every value of the collapse velocity; they exhibit two localized modes that evolve continuously as a function of V from the subsonic to the supersonic regime. These two modes are analytically and numerically shown to merge together in the limit V→∞.