Energy invariant for shallow-water waves and the Korteweg–de Vries equation: Doubts about the invariance of energy

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.

Figure 1

Schematic view of the geometry.

Figure 2

Precision of energy conservation for three-soliton solution. Energies are plotted as open circles $\left(E_1\right)$ and open squares $\left(E_2\right)$ for 40 time instants.

Figure 3

Shape evolution of three-soliton solution during collision.

Figure 4

Energy (non-)conservation for the extended KdV equation in the fixed frame (1). Symbols represent values of the total energy given by formulas (91) or (97). Full square symbols represent the invariant $I^{\left(1\right)}$.

Figure 5

Energy (non-)conservation for the extended KdV equation in the moving frame (116). Symbols represent values of the total energy given by the formula (103). Full square symbols represent the invariant $I^{\left(1\right)}$.

Figure 6

Example of time evolution of the three-soliton solution.