On the deep-water and shallow-water limits of the intermediate long wave equation from a statistical viewpoint

We study convergence problems for the intermediate long wave (ILW) equation, with the depth parameter (Formula presented.), in the deep-water limit ((Formula presented.)) and the shallow-water limit ((Formula presented.)) from a statistical point of view. In particular, we establish convergence of invariant Gibbs dynamics for ILW in both the deep-water and shallow-water limits. For this purpose, we first construct the Gibbs measures for ILW, (Formula presented.). As they are supported on distributions, a renormalization is required. With the Wick renormalization, we carry out the construction of the Gibbs measures for ILW. We then prove that the Gibbs measures for ILW converge in total variation to that for the Benjamin–Ono (BO) equation in the deep-water limit ((Formula presented.)). In the shallow-water regime, after applying a scaling transformation, we prove that, as (Formula presented.), the Gibbs measures for the scaled ILW converge weakly to that for the Korteweg–de Vries (KdV) equation. We point out that this second result is of particular interest because the Gibbs measures for the scaled ILW and KdV are mutually singular (whereas the Gibbs measures for ILW and BO are equivalent). In terms of dynamics, we use a compactness argument to construct invariant Gibbs dynamics for ILW (without uniqueness). Furthermore, we show that, by extracting a sequence (Formula presented.), this invariant Gibbs dynamics for ILW converges to that for BO in the deep-water limit ((Formula presented.)) and to that for KdV (after the scaling) in the shallow-water limit ((Formula presented.)), respectively. Finally, we point out that our results also apply to the generalized ILW equation in the defocusing case, converging to the generalized BO in the deep-water limit and to the generalized KdV in the shallow-water limit. In the non-defocusing case, however, our results cannot be extended to a nonlinearity with a higher power due to the nonnormalizability of the corresponding Gibbs measures.