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

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.