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

We consider the generalized Benjamin-Ono (gBO) equation on the real line, u(t) + partial derivative(x)(-Hu(x)+ 1 /m u(m)) = 0, x is an element of R, m = 2, 3, 4, 5, and perform numerical study of its solutions. We first 1 compute the ground state solution to -Q - HQ' + 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 blowup 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. (C) 2021 Elsevier Inc. All rights reserved.