Dynamics of solutions in the generalized Benjamin-Ono equation: a numerical study

TL;DR

This paper numerically studies the generalized Benjamin-Ono equation, exploring soliton behavior, blow-up phenomena, and solution dynamics across different criticalities, confirming several conjectures and revealing new insights into solution behavior.

Contribution

It provides the first detailed numerical analysis of the gBO equation across various critical regimes, confirming soliton resolution, blow-up, and global existence conjectures.

Findings

Confirmation of soliton resolution in the Benjamin-Ono case.

Observation of stable blow-up near the ground state in the mBO case.

Numerical evidence supporting the blow-up vs global existence dichotomy in supercritical cases.

Abstract

We consider the generalized Benjamin-Ono (gBO) equation on the real line, $ u_t + \partial_x (-\mathcal H u_{x} + \tfrac1{m} u^m) = 0, x \in \mathbb R, m = 2,3,4,5$, and perform numerical study of its solutions. We first compute the ground state solution to $-Q -\mathcal H Q^\prime +\frac1{m} Q^m = 0$ via Petviashvili's iteration method. We then investigate the behavior of solutions in the Benjamin-Ono ($m=2$) equation for initial data with different decay rates and show decoupling of the solution into a soliton and radiation, thus, providing confirmation to the soliton resolution conjecture in that equation. In the mBO equation ($m=3$), which is $L^2$-critical, we investigate solutions close to the ground state mass, and, in particular, we observe the formation of stable blow-up above it. Finally, we focus on the $L^2$-supercritical gBO equation with $m=4,5$. In that case we investigate the global vs finite time existence of solutions, and give numerical confirmation for the dichotomy conjecture, in particular, exhibiting blow-up phenomena in the supercritical setting.