Peakons

30 Oct 2016

I apologize for a long break from the last post. However, it does not mean that we did not do anything :) On the contrary!

The first preprint is devoted to the famous Whitham equation as a model of long capilllary-gravity waves. It is not available yet on Arxiv, but it can be found, for example, in ResearchGate or I can send it back after a simple request by e-mail:

• E. Dinvay, D. Moldabayev, D. Dutykh & H.Kalisch. The Whitham equation with surface tension, Submitted, 2016
• Abstract: The Whitham equation was proposed as an alternate model equation for the simplified description of unidirectional wave motion at the surface of an inviscid fluid. As the Whitham equation incorporates the full linear dispersion relation of the water wave problem, it is thought to provide a more faithful description of shorter waves of small amplitude than traditional long wave models such as the KdV equation. In this work, we derive the Whitham equation from the Hamiltonian theory of surface water waves while taking into account surface tension. It is shown numerically that in various scaling regimes the Whitham equation gives a more accurate approximation of the free surface problem for the Euler system than other models like the KdV, BBM or Kawahara equation. Only in the case of very long waves with positive polarity do the KdV and Kawahara equations outperform the Whitham equation with surface tension.

Another preprint was submitted a couple of days ago. It reports our recent findings of peakon-like travelling waves to capillary-gravity Serre-Green-Naghdi equations in the critical regime:

• D. Mitsotakis, D. Dutykh, A. Assylbekuly & D. Zhakebaev. On weakly singular and fully nonlinear travelling shallow capillary-gravity waves in the critical regime, Submitted, 2016
• Abstract: In this Letter we consider long capillary-gravity waves described by a fully nonlinear weakly dispersive model. First, using the phase space analysis methods we describe all possible types of localized travelling waves. Then, we especially focus on the critical regime, where the surface tension is exactly balanced by the gravity force. We show that our long wave model with a critical Bond number admits stable travelling wave solutions with a singular crest. These solutions are usually referred to in the literature as peakons or peaked solitary waves. They satisfy the usual speed-amplitude relation, which coincides with Scott-Russel’s empirical formula for solitary waves, while their decay rate is the same regardless their amplitude. Moreover, they can be of depression or elevation type independent of their speed. The dynamics of these solutions are studied as well.

Let us hope they will be published soon!

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