Instability of solitons and collapse of acoustic waves in media with positive dispersion

TL;DR

This paper reviews the instability and collapse of acoustic waves in media with positive dispersion, analyzed through the three-dimensional KP equation, highlighting the self-focusing instability and collapse criteria supported by numerical and analytical methods.

Contribution

It presents new insights into the KP instability of multidimensional solitons and the collapse phenomena in acoustic media with positive dispersion, using variational and quasiclassical approaches.

Findings

KP instability occurs for 1D and 2D solitons in 3D media.

Collapse is linked to Hamiltonian unboundedness from below.

Self-similar collapse solutions are identified.

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.