Tadahiro Oh, University of Edinburgh
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 the deterministic setting, convergence problems for ILW have been studied and it is known that, under appropriate assumptions, ILW converges to BO in the deep-water limit and KdV in the shallow-water limit. In this talk, I will discuss convergence problems for ILW from a statistical viewpoint.
In the first part of the talk, I will talk about convergence of the Gibbs measures for ILW and their associated dynamics, where modes of convergence of the Gibbs measures in the deep water and shallow-water limits are different.
In the second part of the talk, I will exploit complete integrability of ILW (and also of BO and KdV) and discuss convergence of higher order conservation laws for ILW and their associated measures. In particular, KdV, appearing in the shallow-water limit, possesses half as many conservation laws as ILW and BO and thus there is an interesting 2-to-1 collapse of ILW conservation laws to those of KdV, which yields alternating modes of convergence of the associated measures in the shallow-water regime.
This talk is based on joint works with Guopeng Li, Andreia Chapouto (both Edinburgh) and Guangqu Zheng (Liverpool).