title: Energy invariant for shallow-water waves and the Korteweg-de Vries equation: Doubts about the invariance of energy.
abstract: It is well known that the Korteweg-de Vries (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 within the accuracy of the expansion.
authors.affiliation: Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra, Poland.
authors.forename: Anna
authors.initials: A
authors.lastname: Karczewska
authors.affiliation: Institute of Physics, Faculty of Physics and Astronomy University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra, Poland.
authors.forename: Piotr
authors.initials: P
authors.lastname: Rozmej
authors.affiliation: National Centre for Nuclear Research, Hoża 69, 00-681 Warszawa, Poland.
authors.forename: Eryk
authors.initials: E
authors.lastname: Infeld