Energy invariant for shallow water waves and the Korteweg -- de Vries equation. Is energy always an invariant?

Abstract

It is well known that the KdV equation has an infinite set of conserved quantities. The first three are often considered to represent mass, momentum and energy. Here we try to answer the question of how this comes about, and also how these KdV quantities relate to those of the Euler shallow water equations. Here Luke's Lagrangian is helpful. We also consider higher order extensions of KdV. Though in general not integrable, in some sense they are almost so.