The intermediate long wave equation (ILW) models water waves of finite depth, connecting the Benjamin-Ono equation (deep-water limit) and the KdV equation (shallow-water limit). Convergence problems of ILW (in both the deep-water and shallow-water limits) have attracted attention from both the applied and theoretical points of view. In this talk, I will discuss convergence problems from a statistical viewpoint.
I first consider the convergence problem of the Gibbsian ensembles. In this case, I establish convergence of the Gibbs measures and then also show convergence of invariant Gibbs dynamics to that of the Benjamin-Ono and KdV equations (without uniqueness). ILW is known to be completely integrable and thus possesses infinitely many conservation laws. In the second part of the talk, I consider invariant dynamics for ILW associated with higher order conservation laws. Due to a complicated nature of the dispersion, even the construction of measures associated with higher order conservation laws turns out to be highly non-trivial. By considering a suitable combination of higher order conservation laws, I overcome this issue and construct invariant dynamics for ILW with a fixed depth parameter. In the final part, I will discuss convergence of the invariant dynamics associated with higher order conservation laws.
This talk is based on a joint work with Tadahiro Oh (Edinburgh), Guangqu Zheng (Liverpool), and Andreia Chapouto (UCLA).