We study the nonlinear dynamics of localized perturbations of a confined generic boundary-layer shear flow in the framework of the essentially two-dimensional generalization of the intermediate long-wave (2d-ILW) equation. The 2d-ILW equation was originally derived to describe nonlinear evolution of boundary layer perturbations in a fluid confined between two parallel planes. The distance between the planes is characterized by a dimensionless parameter D. In the limits of large and small D, the 2d-ILW equation respectively tends to the 2d Benjamin-Ono and 2d Zakharov-Kuznetsov equations. We show that localized initial perturbations of any given shape collapse, i.e., blow up in a finite time and form a point singularity, if the Hamiltonian is negative, which occurs if the perturbation amplitude exceeds a certain threshold specific for each particular shape of the initial perturbation. For axisymmetric Gaussian and Lorentzian initial perturbations of amplitude a and width σ, we derive explicit nonlinear neutral stability curves that separate the domains of perturbation collapse and decay on the plane (a, σ) for various values of D. The amplitude threshold a increases as D and σ decrease and tends to infinity at D → 0. The 2d-ILW equation also admits steady axisymmetric solitary wave solutions whose Hamiltonian is always negative; they collapse for all D except D = 0. But the equation itself has not been proved for small D. Direct numerical simulations of the 2d-ILW equation with Gaussian and Lorentzian initial conditions show that initial perturbations with an amplitude exceeding the found threshold collapse in a self-similar manner, while perturbations with a below-threshold amplitude decay.

中文翻译：

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二维中间长波方程描述的边界层坍塌

我们在中间长波 (2d-ILW) 方程的基本二维概括的框架内研究了受限通用边界层剪切流的局部扰动的非线性动力学。2d-ILW 方程最初是用来描述限制在两个平行平面之间的流体中边界层扰动的非线性演化的。平面之间的距离由无量纲参数 D 表征。在大 D 和小 D 的范围内，2d-ILW 方程分别趋于 2d Benjamin-Ono 和 2d Zakharov-Kuznetsov 方程。我们证明了任何给定形状的局部初始扰动崩溃，即在有限时间内爆炸并形成点奇点，如果哈密顿量为负，如果扰动幅度超过特定于初始扰动的每个特定形状的特定阈值，就会发生这种情况。对于振幅 a 和宽度 σ 的轴对称高斯和洛伦兹初始扰动，我们推导出显式非线性中性稳定性曲线，该曲线将平面 (a, σ) 上的扰动塌陷和衰减域分离为不同的 D 值。振幅阈值 a 增加为D 和 σ 减小并在 D → 0 处趋于无穷大。2d-ILW 方程也允许稳定的轴对称孤立波解，其哈密顿量始终为负；对于除 D = 0 之外的所有 D，它们都会崩溃。但是对于小 D，该方程本身尚未得到证明。