Approximate Conservation Laws in the KdV Equation

The Korteweg-de Vries equation is known to yield a valid description of surface waves for waves of small amplitude and large wavelength. The equation features a number of conserved integrals, but there is no consensus among scientists as to the physical meaning of these integrals. In particular, it is not clear whether these integrals are related to the conservation of momentum or energy, and some researchers have questioned the conservation of energy in the dynamics governed by the equation. In this note it is shown that while exact energy conservation may not hold, if momentum and energy densities and fluxes are defined in an appropriate way, then solutions of the Korteweg-de Vries equation give rise to approximate differential balance laws for momentum and energy.