论文标题
孤子子的不稳定性和媒介中的声波崩溃，具有正分散
Instability of solitons and collapse of acoustic waves in media with positive dispersion
论文作者
论文摘要
本文简要回顾了研究媒介中具有正分散的介质中声波崩溃的结果，这是用三维Kadomtsev-Petviashvili（KP）方程来描述的。使用相应光谱问题的膨胀考虑，考虑使用长波长极限的一维孤子子的KP不稳定性。结果表明，在三维KP方程的框架中，KP不稳定也发生在具有正分散的框架中。根据B.B. Kadomtsev的说法，这种不稳定性属于自我关注的类型。这种不稳定性的非线性阶段是崩溃。崩溃标准之一是下面的汉密尔顿无限制，与$ l_2 $ norm相吻合的固定动量投影。这一事实来自缩放转换，使该规范不变。因此，崩溃可以表示为在自洽的无限电位中掉落到中心的过程。结果表明，由于其不受限制的无限性，波浪的辐射促进了波浪的塌陷。数值实验\ cite {kuznetsovmushershafarenko1983，kuznetsovmusher1986}证实了这种情况。介绍了两种分析方法：使用变异方法和准经典近似。与具有聚焦非线性的非线性Schrödinger方程（NLSE）相反，描述声学崩溃的准经典方法的特征是，该方法是针对三维KP方程作为流体动力非线性的系统提出的。在准经典描述的框架内，发现了一个自相似的家族。
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 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.
