Higher dimensional generalization of the Benjamin-Ono equation: 2D case

TL;DR

This paper studies a 2D higher-dimensional Benjamin-Ono equation, analyzing solution properties both analytically and numerically, including existence, bounds, and wave interactions, revealing critical thresholds for solution behavior.

Contribution

It introduces a numerical approach to the 2D critical HBO equation, derives non-existence conditions for solitary waves, and explores solution dynamics and interactions in a higher-dimensional setting.

Findings

Ground state mass acts as a threshold for global existence.

Globally existing solutions disperse into radiation.

Blow-up solutions travel with rescaled ground state profile.

Abstract

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.