Exact solitonic and periodic solutions of the extended KdV equation

TL;DR

This paper derives exact solitonic and periodic solutions for the extended KdV equation, including higher order effects and uneven bottoms, revealing specific parameter constraints and solution behaviors.

Contribution

It provides explicit solutions for the extended KdV equation with higher order effects and uneven bottoms, highlighting conditions for solitons and cnoidal waves.

Findings

Exact solutions exist for specific parameter ranges.

KdV2 solutions require particular amplitude-to-depth ratios.

Cnoidal waves invert near zero elliptic parameter m.

Abstract

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 (KdV2) and an uneven river bottom. Although this equation is not integrable and has only one conservation law, exact periodic and solitonic solutions exist for the even bottom case. The method used to find them assumes the same function forms as for KdV solutions. KdV2 equation imposes more constraints on parameters of solutions. For soliton case KdV2 solution occurs for particular ratio of wave amplitude to water depth only. For periodic case physically relevant solutions are admissible only for two narrow intervals of elliptic parameter $m$. For a range of $m$ near one the cnoidal waves are upright as expected, but are inverted in $m$ region close to zero. Properties of exact solutions of KdV and KdV2 are compared.