A bound on oscillations in an unsteady undular bore

Abstract

Using a model equation for the evolution of long waves at the surface of an incompressible fluid, the number of rapid oscillations of an undular bore is estimated. This estimate relies on the analysis of the Burgers–Korteweg–deVries equation in spaces of analytic functions. Special attention is paid to the effect of viscosity.

Keywords:

• undular bore
• KdV-Burgers equation
• Jensen's formula
• counting oscillations
• radius of analyticity

AMS Subject Classifications::

• 35Q53
• 76B15
• 76D33

Acknowledgements

This work was supported in part by the Research Council of Norway. The authors also wish to thank two anonymous referees whose comments helped to improve the exposition of the article significantly.