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

摘要

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