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

TL;DR

This work analyzes finite-time collapses of localized perturbations in the essentially two-dimensional Benjamin-Ono (2D-BO) equation derived from Navier–Stokes. Using pseudospectral simulations, the authors demonstrate threshold-driven collapses (when the Hamiltonian $H<0$) and show that, near the singularity, the evolving pulse becomes axially symmetric and self-similar with amplitude scaling $( ext{check}{\tau})^{-\lambda}$ and width scaling $(\text{check}{\tau})^{\lambda}$, where $\lambda \approx 0.9$. Two self-similar solution families are developed: a general anisotropic one and an axially symmetric one, with the latter reducing to a 1D integral equation whose radial Benjamin-Ono soliton fits the numerical collapse profiles. The results reveal a robust, nearly universal collapse mechanism in a 2D nonaxially symmetric dispersive system and establish a one-dimensional near-singularity model governed by Townes-like soliton dynamics.

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 τ)^{-λ}$ and its transverse scale shrinking as $ (\check τ)^λ$, where $\check τ$ 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 $ λ$ 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: $ λ \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.

Figures (7)

• Figure 1: Sketch of geometry of a typical free-surface boundary layer profile. There are no assumptions regarding the profile. Usually, free-surface boundary layer has the maximum of velocity at the surface $U_{max}=U(0)$ but this point is immaterial for the study.
• Figure 2: Snapshots showing the evolution of the amplitude $A(x,y,\tau)$ of a collapsing initially laterally stretched ($\alpha=2$) Gaussian pulse taken at four-different moments: $\tau=0, \, 40, \,70, \,100$. The parameters of the initial Gaussian pulse prescribed by (7) ($a=0.3159, \sigma_{x}=25, \sigma_{y}=50$).
• Figure 3: Evolution a collapsing initially laterally stretched Gaussian pulse ($\alpha=2$) taken at six-different moments: cross-sections. $\sigma_{x}=25,\,\sigma_{y}=50,\,a=0.1353$
• Figure 4: Examples of time evolution of the amplitude of a collapsing pulse with an initially Gaussian distribution laterally stretched with the aspect ratio $\alpha=2$ (Solid red line) and longitudinally stretched with $\alpha=\frac{1}{2}$ (Black solid line). Plot shows nondimensional amplitude $A$ against 'slow time' $\tau, \,t=\varepsilon^{2}\tau$. Red solid line: simulated evolution of the amplitude of a collapsing pulse with the initial condition ($a=0.1353,\sigma_{x}=25, \sigma_{y}=50$). Black solid line: simulated evolution of the amplitude of a collapsing pulse with the initial condition ($a=0.1353,\sigma_{x}=50, \sigma_{y}=25$).
• Figure 5: Exact numerical solution of \ref{['eqnn-spatial-symmetric']} (blue dashed line) and an example of the 'radial' BO soliton approximation in $\xi$ variable (solid red line) and the superimposed upon it. %It shows how well the Lorentzian pulse BO solution captures