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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jüri Engelbrecht What physical effects are involved?, Questions About Elastic Waves (2015), p. 91 | DOI:10.1007/978-3-319-14791-8_6
Andrus Salupere; Martin Lints; Jüri Engelbrecht On solitons in media modelled by the hierarchical KdV equation, Archive of Applied Mechanics, Volume 84 (2014) no. 9-11, pp. 1583-1593 | DOI:10.1007/s00419-014-0861-y | Zbl:1341.35145
Jüri Engelbrecht; Andrus Salupere Scaling and hierarchies of wave motion in solids, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, Volume 94 (2014) no. 9, pp. 775-783 | DOI:10.1002/zamm.201300042 | Zbl:1298.74125
A. Berezovski; J. Engelbrecht; A. Salupere; K. Tamm; T. Peets; M. Berezovski Dispersive waves in microstructured solids, International Journal of Solids and Structures, Volume 50 (2013) no. 11-12, p. 1981 | DOI:10.1016/j.ijsolstr.2013.02.018
Hongwei Wang; Amin Esfahani Well-posedness for the Cauchy problem associated to a periodic Boussinesq equation, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 89 (2013), pp. 267-275 | DOI:10.1016/j.na.2013.04.011 | Zbl:1284.35353
Igor V. Andrianov; Vladyslav V. Danishevs’kyy; Oleksandr I. Ryzhkov; Dieter Weichert Dynamic homogenization and wave propagation in a nonlinear 1D composite material, Wave Motion, Volume 50 (2013) no. 2, p. 271 | DOI:10.1016/j.wavemoti.2012.08.013
Andrus Salupere; Kert Tamm On the influence of material properties on the wave propagation in Mindlin-type microstructured solids, Wave Motion, Volume 50 (2013) no. 7, pp. 1127-1139 | DOI:10.1016/j.wavemoti.2013.05.004 | Zbl:1454.74097
Ramon Quintanilla; Giuseppe Saccomandi Phragmén-Lindelöf alternative of exponential type for the solutions of a fourth order dispersive equation, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni, Volume 23 (2012) no. 2, pp. 105-113 | DOI:10.4171/rlm/620 | Zbl:1245.35127
A.V. Porubov; I.V. Andrianov; V.V. Danishevs’kyy Nonlinear strain wave localization in periodic composites, International Journal of Solids and Structures, Volume 49 (2012) no. 23-24, p. 3381 | DOI:10.1016/j.ijsolstr.2012.07.008
Amin Esfahani; Steven Levandosky Stability of solitary waves for the generalized higher-order Boussinesq equation, Journal of Dynamics and Differential Equations, Volume 24 (2012) no. 2, pp. 391-425 | DOI:10.1007/s10884-012-9250-9 | Zbl:1260.35180
Christo I. Christov Numerical implementation of the asymptotic boundary conditions for steadily propagating 2D solitons of Boussinesq type equations, Mathematics and Computers in Simulation, Volume 82 (2012) no. 6, pp. 1079-1092 | DOI:10.1016/j.matcom.2010.07.030 | Zbl:1320.76031
Kert Tamm; Andrus Salupere On the propagation of 1D solitary waves in Mindlin-type microstructured solids, Mathematics and Computers in Simulation, Volume 82 (2012) no. 7, pp. 1308-1320 | DOI:10.1016/j.matcom.2010.06.022 | Zbl:1348.74181
C. I. Christov; J. Choudhury Perturbation solution for the 2D Boussinesq equation, Mechanics Research Communications, Volume 38 (2011) no. 3, pp. 274-281 | DOI:10.1016/j.mechrescom.2011.01.014 | Zbl:1272.76182
Gérard A. Maugin Solitons in elastic solids (1938–2010), Mechanics Research Communications, Volume 38 (2011) no. 5, pp. 341-349 | DOI:10.1016/j.mechrescom.2011.04.009 | Zbl:1272.74364
Arkadi Berezovski; Jüri Engelbrecht; Mihhail Berezovski Dispersive Wave Equations for Solids with Microstructure, Vibration Problems ICOVP 2011, Volume 139 (2011), p. 699 | DOI:10.1007/978-94-007-2069-5_94
Jüri Engelbrecht; Andrus Salupere; Kert Tamm Waves in microstructured solids and the Boussinesq paradigm, Wave Motion, Volume 48 (2011) no. 8, pp. 717-726 | DOI:10.1016/j.wavemoti.2011.04.001 | Zbl:1365.74139
Gérard A. Maugin Generalized continuum mechanics: what do we mean by that?, Mechanics of generalized continua. One hundred years after the Cosserats. Papers based on the presentations at the EUROMECH colloquium 510, Paris, France, May 13–16, 2009., New York, NY: Springer, 2010, pp. 3-13 | DOI:10.1007/978-1-4419-5695-8_1 | Zbl:1396.74023
C. I. Christov The concept of a quasi-particle and the non-probabilistic interpretation of wave mechanics, Mathematics and Computers in Simulation, Volume 80 (2009) no. 1, pp. 91-101 | DOI:10.1016/j.matcom.2009.06.015 | Zbl:1406.74362

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