What kinds of KdV-type equations are allowed by an uneven bottom

Abstract

In this study, we give a survey of derivations of KdV-type equations with an uneven bottom for several cases when small (perturbation) parameters $\alpha ,\beta ,\delta$ are of different orders. Six different cases of such ordering are discussed. Surprisingly, for all these cases the Boussinesq equations can be made compatible only for the particular piecewise linear bottom profiles, and the correction function has a universal form. For such bottom relief, several new KdV-type wave equations are derived. These equations generalize the KdV, the extended KdV (KdV2), the fifth-order KdV (KdV5) and the Gardner equations.

Mistakes corrected. 24 pages, 5 figures, 1 table

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Cited by in corpus (6)

• (2+1)-dimensional KdV, fifth-order KdV, and Gardner equations derived from the ideal fluid model. Soliton, cnoidal and superposition solutions
• Model order reduction strategies for weakly dispersive waves
• Boussinesq's equations for (2+1)-dimensional gravity-surface waves in an ideal fluid model
• Comment on "Two-dimensional third- and fifth-order nonlinear evolution equations for shallow water waves with surface tension" [Nonlinear Dyn, doi:10.1007/s11071-017-3938-7]
• Inverted solutions of KdV-type and Gardner equations
• Generalized KdV-type equations versus Boussinesq's equations for uneven bottom -- numerical study