Benjamin-Ono Soliton Dynamics in a Slowly Varying Potential

We consider the Benjamin-Ono equation with a slowly varying potential $u_t + (Hu_x-Vu + \tfrac12 u^2)_x=0$ 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 $Q_{a,c}(x) = cQ(c(x-a))$, 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 in $H_x^{1/2}$ to $Q_{0,1}(x)$, we show that the solution $u(x,t)$ to (pBO) remains close in $H_x^{1/2}$ to $Q_{a(t),c(t)}(x)$ and specify the $(a,c)$ parameter dynamics on an $O(h^{-1})$ time scale.