The intermediate long wave equation (ILW) models the internal wave propagation of the interface in a stratified fluid of finite depth, providing a natural connection between two famous water-wave equations: the Benjamin-Ono equation (= the deep-water regime) and the Korteweg-de Vries equation (= the shallow-water regime).

I will discuss how to exploit the completely integrable nature of the equation to establish the statistical convergence of ILW to both BO and KdV, namely, the convergence of the higher order conservation laws for ILW and of their associated invariant measures. Here, we observe a stark contrast between the two regimes, with distinct modes of convergence as well as a novel 2-to-1 collapse of the invariant ILW dynamics to the corresponding invariant KdV dynamics.