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 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 $L_2$-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 \cite{KuznetsovMusherShafarenko1983, KuznetsovMusher1986}. Two analytical approaches to the study of collapse are presented: using the variational method and the quasiclassical approximation. In contrast to the nonlinear Schr\"odinger 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.