Acta Applicandae Mathematicae

, Volume 133, Issue 1, pp 113-131

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

On the Formulation of Mass, Momentum and Energy Conservation in the KdV Equation

  • Alfatih AliAffiliated withDepartment of Mathematics, University of Bergen
  • , Henrik KalischAffiliated withDepartment of Mathematics, University of Bergen Email author 


The Korteweg-de Vries (KdV) equation is widely recognized as a simple model for unidirectional weakly nonlinear dispersive waves on the surface of a shallow body of fluid. While solutions of the KdV equation describe the shape of the free surface, information about the underlying fluid flow is encoded into the derivation of the equation, and the present article focuses on the formulation of mass, momentum and energy balance laws in the context of the KdV approximation. The densities and the associated fluxes appearing in these balance laws are given in terms of the principal unknown variable η representing the deflection of the free surface from rest position. The formulae are validated by comparison with previous work on the steady KdV equation. In particular, the mass flux, total head and momentum flux in the current context are compared to the quantities Q, R and S used in the work of Benjamin and Lighthill (Proc. R. Soc. Lond. A 224:448–460, 1954) on cnoidal waves and undular bores.


KdV equation Surface waves Mechanical balance laws Energy conservation Hydraulic head