  We consider a higher-dimensional version of the Benjamin-Ono (HBO) equation
in the 2D setting: $u_t- \mathcal{R}_1 \Delta u + \frac{1}{2}(u^2)_x=0, (x,y)
\in \mathbb{R}^2$, which is $L^2$-critical, and investigate properties of
solutions both analytically and numerically. For a generalized equation
(fractional 2D gKdV) after deriving the Pohozaev identities, we obtain
non-existence conditions for solitary wave solutions, then prove uniform bounds
in the energy space or conditional global existence, and investigate the
radiation region, a specific wedge in the negative $x$-direction. We then
introduce our numerical approach in a general context, and apply it to obtain
the ground state solution in the 2D critical HBO equation, then show that its
mass is a threshold for global vs. finite time existing solutions, which is
typical in the focusing (mass-)critical dispersive equations. We also observe
that globally existing solutions tend to disperse completely into the radiation
in this nonlocal equation. The blow-up solutions travel in the positive
$x$-direction with the rescaled ground state profile while also radiating
dispersive oscillations into the radiative wedge. We conclude with examples of
different interactions of two solitary wave solutions, including weak and
strong interactions.
