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

Seminar| Institute of Mathematical Sciences

Abstract: The intermediate long wave equation (ILW) models the internal wave propagation of the interface in a stratified fluid of finite depth, providing a natural connection between the deep-water regime (= the Benjamin-Ono (BO) regime) and the shallow-water regime (= the KdV regime). In this talk, I will discuss convergence problems for ILW from a statistical viewpoint, at different energy levels.



By exploiting both Hamiltonian and completely integrable structure of ILW (and also of BO and KdV), I will discuss convergence of Hamiltonian/higher order conservation laws for ILW, their associated measures and their associated dynamics. In particular, two interesting phenomenons arise: (i) modes of convergence of the measures in the deep water and shallow-water limits are different. (ii) KdV, appearing in the shallow-water limit, possesses half as many conservation laws as ILW and BO leading to a 2-to-1 collapse phenomenon.

This talk is based on joint works with Tadahiro Oh, Andreia Chapouto (both Edinburgh) and Guangqu Zheng (Liverpool).