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(Due to the limit on the number of characters for an abstract set by arXiv,the full abstract can not be displayed here. See the abstract in the paper.) Westudy convergence problems for the intermediate long wave equation (ILW), withthe depth parameter $\delta > 0$, in the deep-water limit ($\delta \to \infty$)and the shallow-water limit ($\delta \to 0$) from a statistical point of view.In particular, we establish convergence of invariant Gibbs dynamics for ILW inboth the deep-water and shallow-water limits. For this purpose, we firstconstruct the Gibbs measures for ILW, $0 < \delta < \infty$. As they aresupported on distributions, a renormalization is required. With the Wickrenormalization, we carry out the construction of the Gibbs measures for ILW.We then prove that the Gibbs measures for ILW converge in total variation tothat for the Benjamin-Ono equation (BO) in the deep-water limit. In theshallow-water regime, after applying a scaling transformation, we prove that,as $\delta \to 0$, the Gibbs measures for the scaled ILW converge weakly tothat for the Korteweg-de Vries equation (KdV). We point out that this secondresult is of particular interest since the Gibbs measures for the scaled ILWand KdV are mutually singular (whereas the Gibbs measures for ILW and BO areequivalent).  We also discuss convergence of the associated dynamical problem.  Lastly, we point out that our results also apply to the generalized ILWequation in the defocusing case, converging to the generalized BO in thedeep-water limit and to the generalized KdV in the shallow-water limit. In thenon-defocusing case, however, our results can not be extended to a nonlinearitywith a higher power due to the non-normalizability of the corresponding Gibbsmeasures.

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