Benjamin-Ono Soliton Dynamics in a slowly varying potential revisited

Abstract

The Benjamin Ono equation with a slowly varying potential is (pBO) ), and H denotes the Hilbert transform. The soliton profile isand a ∈ R, c > 0 are parameters. For initial condition u 0 (x) to (pBO) close to Q 0,1 (x), it was shown in Zhang [48] that the solution u(x, t) to (pBO) remains close to Q a(t),c(t) (x) and approximate parameter dynamics for (a, c) were provided, on a dynamically relevant time scale. In this paper, we prove exact (a, c) parameter dynamics. This is achieved using the basic framework of the paper [48] but adding a local virial estimate for the linearization of (pBO) around the soliton. This is a localin-space estimate averaged in time, often called a local smoothing estimate, showing that effectively the remainder function in the perturbation analysis is smaller near the soliton than globally in space. A weaker version of this estimate is proved in Kenig & Martel [20] as part of a "linear Liouville" result, and we have adapted and extended their proof for our application.