Deep-water and shallow-water limits of statistical equilibria for the intermediate long wave equation

Abstract

We study the construction of invariant measures associated with higher order conservation laws of the intermediate long wave equation (ILW) and their convergence properties in the deep-water and shallow-water limits. By exploiting its complete integrability, we first carry out detailed analysis of the construction of appropriate conservation laws of ILW at the $H^{k/2}$-level for each $k\in \mathbb{N}$, and establish their convergence to those of the Benjamin–Ono equation (BO) in the deep-water limit and to those of the Korteweg–de Vries equation (KdV) in the shallow-water limit. In particular, in the shallow-water limit, we prove a rather striking 2-to-1 collapse of the conservation laws of ILW to those of KdV. Such a 2-to-1 collapse is novel in the literature and, to our knowledge, this is the first construction of a complete family of shallow-water conservation laws with non-trivial shallow-water limits. We then construct an infinite sequence of generalized Gibbs measures for ILW associated with these conservation laws and prove their convergence to the corresponding (invariant) generalized Gibbs measures for BO and KdV in the respective limits. Finally, for $k\ge 3$, we establish invariance of these measures under ILW dynamics, and also convergence in the respective limits of the ILW dynamics at each equilibrium state to the corresponding invariant dynamics for BO and KdV constructed by Deng, Tzvetkov, and Visciglia (2010–2015) and Zhidkov (1996), respectively. In particular, in the shallow-water limit, we establish a 2-to-1 collapse at the level of the generalized Gibbs measures as well as the invariant ILW dynamics. As a byproduct of our analysis, we also prove invariance of the generalized Gibbs measure associated with the $H^2$-conservation law of KdV, which seems to be missing in the literature.