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

Abstract We study convergence problems for the intermediate long wave (ILW) equation, with the depth parameter , in the deep‐water limit () and the shallow‐water limit () 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, . 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 (). In the shallow‐water regime, after applying a scaling transformation, we prove that, as , 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 , this invariant Gibbs dynamics for ILW converges to that for BO in the deep‐water limit () and to that for KdV (after the scaling) in the shallow‐water limit (), 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.