Instability of Solitons and Collapse of Acoustic Waves in Media with Positive Dispersion

Abstract

This article is a brief review of the results of studying the collapse of sound waves in media with positive dispersion, which is described in terms of the three-dimensional Kadomtsev–Petviashvili (KP) equation. The KP instability of one-dimensional solitons in the long-wavelength limit is considered using the expansion for the corresponding spectral problem. It is shown that the KP instability also takes place for two-dimensional solitons in the framework of the three-dimensional KP equation with positive dispersion. According to Kadomtsev (see Collective Effects in Plasma by B.B. Kadomtsev) this instability belongs to the self-focusing type. The nonlinear stage of this instability is a collapse. One of the collapse criteria is the Hamiltonian unboundedness from below for a fixed momentum projection coinciding with the L₂-norm. This fact follows from scaling transformations, leaving this norm constant. For this reason, collapse can be represented as the process of falling a particle to the center in a self-consistent unbounded potential. It is shown that the radiation of waves from a region with a negative Hamiltonian, due to its unboundedness from below, promotes the collapse of the waves. This scenario was confirmed by numerical experiments. Two analytical approaches to the study of collapse are presented: using the variational method and the quasiclassical approximation. In contrast to the nonlinear Schrödinger equation (NLSE) with a focusing nonlinearity, a feature of the quasiclassical approach to describing acoustic collapse is that this method is proposed for the three-dimensional KP equation as a system with hydrodynamic nonlinearity. Within the framework of the quasiclassical description, a family of self-similar collapses is found. The upper bound of this family corresponds to a strong collapse, in which the energy captures into the singularity is finite. The existence of such a regime is also confirmed based on the variational approach. The other boundary of the collapsing hierarchy coincides with the self-similar solution of the three-dimensional KP equation, which describes the fastest weak collapse.