On Whitham and Related Equations

Abstract

Abstract. The aim of this paper is to study, via theoretical analysis and numerical simulations, the dynamics of Whitham and related equations. In particular we establish rigorous bounds between solutions of the Whitham and KdV equations and provide some insights into the dynamics of the Whitham equation in different regimes, some of them being outside the range of validity of the Whitham equation as a water waves model.

Solitary Wave Solutions to a Class of Modified Green–Naghdi Systems

“…Any other choice satisfying F(k) = 1 + O(k 2 ) enjoys the same precision (in the sense of consistency) in the shallow-water regime and the specific choice of F(k) = 3 d|k| tanh(d|k|) − 3 d 2 |k| 2 allows to obtain a model whose linearization around constant states fits exactly with the one of the water waves system. Hence system (1.1) with the aforementioned choice of Fourier multipliers participates to the recent effort in providing long wave models with the full dispersion property; see [1,11,28,30,39]. However, notice that contrarily to the so-called Boussinesq-Whitham equations, the validity of (1.1) does not rely on any small-amplitude assumption.…”

On the local well-posedness for a full-dispersion Boussinesq system with surface tension

“…In particular, it displays, in the case of pure gravity waves (β = 0), several interesting phenomena already predicted by Whitham: a solitary wave regime close to KdV [9], the existence of a wave of greatest height (Stokes wave) [11], the existence of shocks [13], and modulational instability of steady periodic waves [15,26]. Note that when surface tension is taken into account (β > 0), the dynamics of (1.4) appears to be completely different (see [19] and the references therein). Moreover, it was proved to be a relevant water wave model in the long wave regime on the same time scale as the KdV equation [21,19].…”

Solitary wave solutions to a class of Whitham–Boussinesq systems

“…The BDW system was formally derived in [1,21] from the incompressible Euler equations to model fully dispersive shallow water waves whose propagation is allowed to be both left-and rightward, and appeared in [19,22] as a full dispersion system in the Boussinesq regime with the dispersion of the water waves system. There have been several investigations on the BDW system: local well-posedness [13,18] (in homogeneous Sobolev spaces at a positive background), a logarithmically cusped wave of greatest height [11]. There are also numerical results, investigating the validity of the BDW system as a model of waves on shallow water [4], numerical bifurcation and spectral stability [5] and the observation of dispersive shock waves [24].…”