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

12/06/2020
by   Svetlana Roudenko, et al.
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We consider the generalized Benjamin-Ono (gBO) equation on the real line, u_t + ∂_x (-ℋ u_x + 1m u^m) = 0, x ∈ℝ, m = 2,3,4,5, and perform numerical study of its solutions. We first compute the ground state solution to -Q -ℋ Q^' +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.

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