Deep- and shallow-water convergence of the generalized Gibbs measures for the intermediate long wave equation

This note is based on a talk given by the first author at the conference Journées Équations aux dérivées partielles 2025. We consider the intermediate long wave equation (ILW), modeling the internal wave propagation of the interface in a stratified fluid of finite depth, connecting the deep-water regime (= the BO regime) and the shallow-water regime (= the KdV regime). Exploiting the complete integrability of ILW, we provide a detailed description of its polynomial conservation laws, construct the associated invariant generalized Gibbs measures, and lastly show their convergence to those of BO and KdV. In the shallow-water regime, we establish a novel 2-to-1 collapse of ILW conservation laws to those of KdV (and also a 2-to-1 collapse of the associated generalized Gibbs measures), which exhibits the singular nature of the shallow-water convergence.