Benjamin-Ono Soliton Dynamics in a slowly varying potential revisited

The Benjamin Ono equation with a slowly varying potential is $$\text{(pBO)} \qquad u_t + (Hu_x-Vu + \tfrac12 u^2)_x=0$$ with $V(x)=W(hx)$, $0< h \ll 1$, and $W\in C_c^\infty(\mathbb{R})$, and $H$ denotes the Hilbert transform. The soliton profile is $$Q_{a,c}(x) = cQ(c(x-a)) \,, \text{ where } Q(x) = \frac{4}{1+x^2}$$ and $a\in \mathbb{R}$, $c>0$ are parameters. For initial condition $u_0(x)$ to (pBO) close to $Q_{0,1}(x)$, it was shown in a previous work by Z. Zhang 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 previous work by Z. Zhang but adding a local virial estimate for the linearization of (pBO) around the soliton. This is a local-in-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 a paper by Kenig & Martel as part of a "linear Liouville" result, and we have adapted and extended their proof for our application.