Solutions to the extended KdV equation for water surface waves

Eryk Infeld, Anna Karczewska, George Rowlands, Piotr Rozmej

Published 2016-12-12Version 1

The KdV equation can be derived in the shallow water limit of the Euler equations. Over the last few decades, this equation has been extended to include both higher order effects and an uneven river bottom. Although this equation has only one conservation law, exact periodic and solitonic solutions exist for the even bottom case. The method used to find them assumes a single cnoidal form. Quite unexpectedly, we found two regions in $m$ parameter space. For a range of $m$ near one the cnoidal waves are upright as expected, but are inverted in the remaining $m$ region. A second, more general method which could be applied to other extensions of KdV and indeed other equations is also outlined. This is potentially the other important result of this paper. When the bottom is uneven, our treatment is no longer exact. However, an extremely simple model is obtained in the sequel to this paper. All results are succesfully checked numerically.