On Boussinesq's paradigm in nonlinear wave propagation

Christo I. Christov; Gérard A. Maugin; Alexey V. Porubov

doi:10.1016/j.crme.2007.08.006

Boussinesq equation, elasticity, beams, plates

On Boussinesq's paradigm in nonlinear wave propagation
[Sur la paradigme de Boussinesq pour la propagation d'ondes non linéaires]

Présenté par : Presented by

Christo I. Christov ¹ ; Gérard A. Maugin ² ; Alexey V. Porubov ³

¹ Department of Mathematics, University of Louisiana at Lafayette, P.O. Box 1010, Lafayette, LA 70504-1010, USA
² Université Pierre et Marie Curie, Institut Jean le Rond d'Alembert, UMR 7190, case 162, 4, place Jussieu, 75252 Paris cedex 05, France
³ Institute for Problems in Mechanical Engineering, R.A.S., V.O., Bolshoy avenue 61, Saint-Petersburg 199178, Russia

Comptes Rendus. Mécanique, Joseph Boussinesq, a Scientist of bygone days and present times, Volume 335 (2007) no. 9-10, pp. 521-535.

Résumé
Abstract

L'obtention originale de sa célèbre équation gouvernant les ondes de surface sur une couche fluide par Boussinesq a ouvert de nouveaux horizons qui devaient conduire au concept de soliton. La présente contribution concerne l'ensemble des équations du type Boussinesq sous le titre général de « paradigme de Boussinesq ». Celles-ci sont de véritables équations bi-directionnelles qui apparaissent dans de nombreuses situations physiques et partagent des propriétés analogues. L'accent est mis sur : (i) les système généralisés de Boussinesq qui impliquent une dispersion linéaire d'ordre supérieur soit en raison de la présence de dérivées spatiales d'ordre supérieur, soit avec la contribution d'autres opérateurs d'onde (équation à « double dispersion ») ; et (ii) la « mécanique » des solutions les plus représentatives d'ondes non linéaires localisées qui en résulte. Des généralisations dissipatives et à deux dimensions d'espace sont également envisagées.

Boussinesq's original derivation of his celebrated equation for surface waves on a fluid layer opened up new horizons that were to yield the concept of the soliton. The present contribution concerns the set of Boussinesq-like equations under the general title of ‘Boussinesq's paradigm’. These are true bi-directional wave equations occurring in many physical instances and sharing analogous properties. The emphasis is placed: (i) on generalized Boussinesq systems that involve higher-order linear dispersion through either additional space derivatives or additional wave operators (so-called double-dispersion equations); and (ii) on the ‘mechanics’ of the most representative localized nonlinear wave solutions. Dissipative cases and two-dimensional generalizations are also considered.

Reçu le : 2006-12-11
Accepté le : 2007-03-14
Publié le : 2007-09-01

DOI : 10.1016/j.crme.2007.08.006

Keywords: Computational fluid mechanics, Boussinesq, Nonlinear wave, Elastic solids, Crystal lattice, Solitons, Korteweg–de Vries equation
Mots-clés : Mécanique des fluides numérique, Boussinesq, Ondes non linéaires, Solides élastiques, Réseau cristallin, Solitons, Équation de Korteweg–de Vries

Affiliations des auteurs :

Christo I. Christov ¹ ; Gérard A. Maugin ² ; Alexey V. Porubov ³

¹ Department of Mathematics, University of Louisiana at Lafayette, P.O. Box 1010, Lafayette, LA 70504-1010, USA
² Université Pierre et Marie Curie, Institut Jean le Rond d'Alembert, UMR 7190, case 162, 4, place Jussieu, 75252 Paris cedex 05, France
³ Institute for Problems in Mechanical Engineering, R.A.S., V.O., Bolshoy avenue 61, Saint-Petersburg 199178, Russia

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Christo I. Christov; Gérard A. Maugin; Alexey V. Porubov. On Boussinesq's paradigm in nonlinear wave propagation. Comptes Rendus. Mécanique, Joseph Boussinesq, a Scientist of bygone days and present times, Volume 335 (2007) no. 9-10, pp. 521-535. doi : 10.1016/j.crme.2007.08.006. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.08.006/

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