On the formulation of mass, momentum and energy conservation in the KdV equation. (English) Zbl 1310.35206

Summary: 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 \(\eta\) 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 T. B. Benjamin and M. J. Lighthill [Proc. R. Soc. Lond., Ser. A 224, 448–460 (1954; Zbl 0055.45605)] on cnoidal waves and undular bores.


35Q53 KdV equations (Korteweg-de Vries equations)
35A30 Geometric theory, characteristics, transformations in context of PDEs


Zbl 0055.45605
Full Text: DOI


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