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

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