Boussinesq, Schrödinger and Euler-Korteweg

The aim of this paper of survey nature is to emphasize the links between two specific strongly dispersive “abcd” Boussinesq systems and nonlinear Schrödinger equations, one of those systems being a particular case of the Euler-Korteweg (EK) systems. We will in particular translate results for the later the known results for the EK system. We also comment on a one-dimensional integrable version of those systems, known as the Kaup-Broer-Kupperschmidt system. Finally we discuss another Boussinesq system that can be viewed as a (weakly) dispersive perturbation of the Saint-Venant (shallow-water) system.