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

TL;DR

This paper investigates the collapse dynamics of localized perturbations in the 2D Benjamin-Ono equation, revealing universal self-similar behavior and axial symmetry near singularities, with implications for simplified collapse modeling.

Contribution

It introduces a self-similar solution framework for axially symmetric collapses in the 2D Benjamin-Ono equation, demonstrating near-universality of collapse parameters and a reduction to an effective one-dimensional model.

Findings

Localized perturbations exceeding a threshold collapse into a point singularity.

Near the singularity, solutions become axially symmetric regardless of initial shape.

The collapse exhibits self-similar behavior with a nearly universal parameter λ ≈ 0.9.

Abstract

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.