Benjamin-Ono soliton dynamics in a slowly varying potential

PDE Seminar
Thursday, January 16, 2020 - 11:00am for 1 hour (actually 50 minutes)
Skiles 005
Zhiyuan Zhang – Brown University –
Zhiwu Lin

We consider the Benjamin Ono equation, modeling one-dimensional long interval waves in a stratified fluid, with a slowly-varying potential perturbation. Starting with near soliton initial data, we prove that the solution remains close to a soliton wave form, with parameters of position and scale evolving according to effective ODEs depending on the potential. The result is valid on a time-scale that is dynamically relevant, and highlights the effect of the perturbation. It is proved using a Lyapunov functional built from energy and mass, Taylor expansions, spectral estimates, and estimates for the Hilbert transform.