标题： 二维 Benjamin-Ono 方程中的轴对称塌陷 显示英文标题

标题： Axially symmetric collapses in the 2-D Benjamin-Ono equation

摘要： 我们研究了在本质上二维的Benjamin-Ono方程（2D-BO）框架内局部位移的非线性动力学，该方程是从纳维-斯托克斯方程渐近推导出来的。通过使用拟谱方法模拟2D-BO方程，我们确认超过某一阈值的局部位移初始扰动会塌陷，形成一个点奇异性。 尽管2D-BO方程不具备轴对称性，我们证明在塌陷奇异性附近，无论其初始形状如何，解都变得轴对称。 我们发现扰动以自相似的方式塌陷，扰动振幅爆炸性地增长为$ (\check \tau)^{-\lambda}$，其横向尺度缩小为$ (\check \tau)^{\lambda}$，其中$\check \tau$是到奇异性时刻的时间。 我们推导了一组描述轴对称塌陷的自相似解。 自相似解中的自由参数$ {\lambda}$的值通过将其拟合到初始问题的数值模拟中，即演化开始时局部位移的演变问题中来确定。 值得注意的是，对于所考察的例子，该参数的值几乎具有普遍性：$ {\lambda} \approx 0.9$；它的变化与初始条件无关。 在奇异性附近，动力学变成一维，因此，从2D-BO方程得出的简化提供了一个有效的塌陷一维模型。 显示英文摘要

摘要： We study the nonlinear dynamics of localized perturbations within the framework of the essentially two-dimensional generalization of the Benjamin-Ono equation (2D-BO) derived asymptotically from the Navier-Stokes equation. By simulating the 2D-BO equation with the pseudospectral method, we confirm that the localized initial perturbations exceeding a certain threshold collapse, forming a point singularity. Although the 2D-BO equation does not possess axial symmetry, we show that in the vicinity of the collapse singularity, the solution becomes axially-symmetric, whatever its initial shape. We find that perturbations collapse in a self-similar manner, with the perturbation amplitude exploding as $ (\check \tau)^{-\lambda}$ and its transverse scale shrinking as $ (\check \tau)^{\lambda}$, where $\check \tau$ is the time to the moment of singularity. We derive a family of self-similar solutions describing axially symmetric collapses. The value of the free parameter $ {\lambda}$ in the self-similar solution is specified by fitting it to the numerical simulation of the initial problem of the evolution of an initially localized perturbation. Remarkably, for the examples we examined the value of the parameter proved to be almost universal: $ {\lambda} \approx 0.9$; its dependence on the initial conditions is indiscernible. In the vicinity of the singularity, the dynamics becomes one-dimensional, thus, the derived reduction of the 2D-BO equation provides an effectively one-dimensional model of collapse.