Soliton interaction with slowly varying potentials

We study the Gross-Pitaevskii equation with a slowly varying smooth potential, $V(x) = W(hx)$. We show that up to time $\log(1/h)/h$ and errors of size $h^2$ in $H^1$, the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian, $(\xi^2 + \sech^2 * V (x))/2$. This provides an improvement ($h \to h^2$) compared to previous works, and is strikingly confirmed by numerical simulations.