Denys DUTYKH's Blog of an applied mathematician

Peakons

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!


Latest results

In this post I would like to mention a couple of recent submissions that we prepared with my new and old co-authors.

First of all, with my new brazilian friends (Julien Berger and Nathan Mendes) we submitted a manuscript devoted to the understanding of some inverse problems in the heat conduction (yes, I am totally open to new topics). The preprint is already available on HAL server (it contained too many pictures to :

  • J. Berger, D. Dutykh & N. Mendes. On the optimal experimental design for heat and moisture parameter estimation, Submitted, 2016
  • Abstract: In the context of estimating material properties of porous walls based on in-site measurements and identification method, this paper presents the concept of Optimal Experiment Design (OED). It aims at searching the best experimental conditions in terms of quantity and position of sensors and boundary conditions imposed to the material. These optimal conditions ensure to provide the maximum accuracy of the identification method and thus the estimated parameters. The search of the OED is done using the Fisher information matrix and a priori knowledge on the parameters. The methodology is applied for two cases. The first one deals with purely conductive heat transfer, while the second one combines a strong coupling between heat and moisture transfer.

The second preprint is devoted to my more traditional topics, i.e. water wave modelling. Namely, we consider the deep water case and discuss several models, some of them being well-known and some new. The variational derivations and other averaging techniques are equally considered:

  • D. Dutykh, D. Clamond & M. Chhay. Serre-type equations in deep water, Submitted, 2016
  • Abstract: This manuscript is devoted to the modelling of water waves in the deep water regime with some emphasis on the underlying variational structures. The present article should be considered as a review of some existing models and modelling approaches even if new results are presented as well. Namely, we derive the deep water analogue of the celebrated Serre-Green-Naghdi equations which have become the standard model in shallow water environments. The relation to existing models is discussed. Moreover, the multi-symplectic structure of these equations is reported as well. The results of this work can be used to develop various types of robust structure-preserving variational integrators in deep water. The methodology of constructing approximate models presented in this study can be naturally extrapolated to other physical flow regimes as well.

Now I think it is a good time to recharge the batteries… to undertake new research in September :)

Summer


French-Spanish workshop

Valladolid

Tomorrow I departure for the next destination - Valladolid (Spain) for a French-Spanish Workshop on Evolution Problems (FSWEP16). This meeting will be hosted by imUVa and Universidad de Valladolid. The Programme can be seen here. I would like to thank the organizers for giving me an opportunity to speak there. My talk will be devoted to the derivation of Galilean invariant and energy-consistent long wave models.


Lectures at Curitiba and internal waves

During these two weeks (1st – 14th of April 2016) I will be in Curitiba, Brazil to deliver some lectures on the Numerical Analysis at a PhD school in PUCPR University. I found on YouTube a short video presenting this University:

The Lecture notes are already available. They will be probably updated in the nearest future to the latest version (including some minor corrections and perhaps some new material):

Otherwise, recently with Didier Clamond we submitted a new mauscript where we report the multi-symplectic structure for the Green-Naghdi-type system describing long internal waves evolution. It seems that this system was derived for the first by E. Barthélémy (1989) in his PhD thesis done at the University of Grenoble. For simplicity, we adopt the rigid lid approximation, but the multi-symplectic structure can be generalized to the free surface case as well. The preprint is already freely available to download:

Any comments and suggestions are welcome!


Singular solitary waves

Phase plane analysis

Finally, yesterday with my collaborators from the University of Nice we submitted our manuscript on the construction of singular solitary wave solutions using the classical Phase Plane Analysis (PPA) and some methods from effective Algebraic Geometry.

This work was presented in many seminars from Japan (Yokohama) to Russia (Novosibirsk) and Austria (Linz). So, now you can find all the details about computations in our preprint available at all popular preprints servers: