Ordering of small parameters in nonlinear wave problems

It is a common situation when asymptotic methods are applied to nonlinear wave problems which involve several parameters assumed to be small. As a canonical example, the classical problem of shallow water waves in ideal fluid may be mentioned. In particular, the famous Korteweg–de Vries (KdV) equation, which is the prototypical example of an exactly solvable soliton equation, was first introduced in the context of that problem. The system of equations describing the long, small-amplitude wave motion in shallow water with a free surface involves two independent small parameters: the amplitude parameter α and the wave length parameter β. No relationship between orders of magnitude of α and β follows from the statement of the problem. In the derivation of model equations, the question of ordering is usually not discussed and it is tacitly assumed that the two small parameters are of the same order of magnitude (the derivation of the KdV equation is the case). However, it is evident that there are no grounds for that assumption and that, in general, the parameters α and β can be not of the same order of magnitude. It is indicated in [1], that, in such a case, a consistent truncation of the asymptotic expansion can be made only on the basis of a prescribed relationship between orders of magnitude of α and β, and a systematic procedure for deriving an equation for surface elevation is developed. The results of the analysis provide a set of consistent model equations for unidirectional water waves which replace the KdV equation in the cases of the nonstandard ordering. The problem of shallow water waves over a slowly varying bottom [2], [3] provides an example of the problem which involves three independent small parameters. As other examples of the problems involving several small parameters, the nonlinear interactions among internal oceanic gravity waves and nonlinear instability of (weakly) nonparallel flows are to be considered.

[1] G. I. Burde and A. Sergyeyev, J. Phys. A: Math. Theor. 46, 075501 (2013).
[2] A. Karczewska, P. Rozmej, and E. Infeld, Phys. Rev. E 90, 012907 (2014).
[3] G. I. Burde, Phys. Rev. E 101, 036201 (2020).