Exact solitonic and periodic solutions of the extended KdV equation

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.

8 pages, 6 figures

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Cited by in corpus (9)

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• Martingale solution to stochastic extended Korteweg - de Vries equation
• Remarks on existence/nonexistence of analytic solutions to higher order KdV equations
• Extended KdV equation for the case of uneven bottom
• Inverted solutions of KdV-type and Gardner equations
• Generalized KdV-type equations versus Boussinesq's equations for uneven bottom -- numerical study
• Signatures of chaotic dynamics in wave motion according to the extended KdV equation