Ordering of two small parameters in the shallow water wave problem

TL;DR

This paper develops a systematic asymptotic approach for deriving shallow water wave equations considering different orderings of amplitude and wavelength parameters, revealing new higher-order models and clarifying the emergence of known equations.

Contribution

It introduces a consistent procedure for deriving surface elevation equations with arbitrary parameter orderings, extending the classical KdV framework to include higher-order corrections and alternative model equations.

Findings

Derivation of equations for different parameter orderings, including $eta=O( ext{} ext{alpha}^n)$ and $ ext{alpha}=O(eta^m)$.

Identification of known equations like Gardner, modified KdV, and 5th-order KdV as leading order models.

Development of higher-order models that generalize the KdV equation for non-standard parameter regimes.

Abstract

The classical problem of irrotational long waves on the surface of a shallow layer of an ideal fluid moving under the influence of gravity as well as surface tension is considered. A systematic procedure for deriving an equation for surface elevation for a prescribed relation between the orders of the two expansion parameters, the amplitude parameter $\alpha$ and the long wavelength (or shallowness) parameter $\beta$, is developed. Unlike the heuristic approaches found in the literature, when modifications are made in the equation for surface elevation itself, the procedure starts from the consistently truncated asymptotic expansions for unidirectional waves, a counterpart of the Boussinesq system of equations for the surface elevation and the bottom velocity, from which the leading order and higher order equations for the surface elevation can be obtained by iterations. The relations between the orders of the two small parameters are taken in the form $\beta=O(\alpha^n)$ and $\alpha=O(\beta^m)$ with $n$ and $m$ specified to some important particular cases. The analysis shows, in particular, that some evolution equations, proposed before as model equations in other physical contexts (like the Gardner equation, the modified KdV equation, and the so-called 5th-order KdV equation), can emerge as the leading order equations in the asymptotic expansion for the unidirectional water waves, on equal footing with the KdV equation. The results related to the higher orders of approximation provide a set of consistent higher order model equations for unidirectional water waves which replace the KdV equation with higher-order corrections in the case of non-standard ordering when the parameters $\alpha$ and $\beta$ are not of the same order of magnitude. (See the paper for the complete abstract.)