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 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, these with the accuracy of the expansion.

Single soliton solution to the extended KdV equation over uneven depth

“…However, the full dynamics of the soliton motion is much richer, with the uneven bottom causing low amplitude soliton radiation both ahead and after the main wave. This behaviour was observed in our earlier papers [1,7,8] in which initial conditions were in the form of the KdV soliton, whereas in the present cases the KdV2 soliton, that is, the exact solution of the KdV2 equation has been used.…”

“…This was observed in our earlier papers [1,7,8] in which initial conditions were in the form of the KdV soliton.…”

Symmetries and conservation laws for the Karczewska–Rozmej–Rutkowski–Infeld equation

“…The result of Proposition 1 can be seen as an ultimate amplification of the results of [7] on conservation laws of (1) for constant h.…”

A Fifth-Order Korteweg-de Vries Equation for Shallow Water with Surface Tension: Multiple Soliton Solutions

“…It is interesting to note that the extended KdV equation, as first called by Marchant and Smyth [31], of second order in small parameters, was subjected to useful studies in [32][33][34] and some of the references therein. In Ref.…”

“…Moreover, in [33], the derivation of a KdV type equation, second order in small parameters, containing terms from the bottom function was examined. The energy invariant for shallow-water waves and the Korteweg-de Vries equation were examined in detail in [34]. Generally, KdV2 equation (second order with respect to small parameters) are examined thoroughly in [31][32][33][34] in addition to [1,2].…”