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

TL;DR

This paper investigates the invariance of energy in shallow water wave models, particularly the Korteweg--de Vries equation, exploring how conserved quantities relate to physical properties and extending to higher-order, nearly integrable models.

Contribution

It clarifies the relationship between KdV conserved quantities and physical invariants in shallow water equations, and examines higher-order extensions for near-integrability.

Findings

Energy is not always an invariant in extended models.

The first three KdV invariants relate to physical quantities like mass, momentum, and energy.

Higher-order extensions are nearly integrable but not exactly so.

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.