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

Abstract

We consider the generalized Benjamin-Ono (gBO) equation on the real line, $u_t+{\partial }_x(-\mathcal{H}{u}_x+\frac{1}{m}{u}^m)=0,x\in \mathbb{R},m=2,3,4,5$, and perform numerical study of its solutions. We first compute the ground state solution to $-Q-\mathcal{H}{Q}^{\prime }+\frac{1}{m}{Q}^m=0$ 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.