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, 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
Mot 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

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Christo I. Christov; Gérard A. Maugin; Alexey V. Porubov. On Boussinesq's paradigm in nonlinear wave propagation. Comptes Rendus. Mécanique, 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/

Bibliographie
Cité par

[1] J.V. Boussinesq Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire, C. R. Acad. Sci. Paris, Volume 72 (1871), pp. 755-759

[2] J.V. Boussinesq Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., Volume 17 (1872), pp. 55-108

[3] Lord Rayleigh On waves, Phil. Mag. (5), Volume 1 (1876), pp. 257-279

[4] D.J. Korteweg; G. de Vries On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag. (5), Volume 39 (1895), pp. 422-443

[5] F. Ursell The long-wave paradox in the theory of gravity waves, Proc. Cambr. Phil. Soc., Volume 49 (1953), pp. 685-694

[6] C.I. Christov Conservative difference scheme for Boussinesq model of surface waves (K. Morton; J. Baines, eds.), Proceedings ICFD 5, Oxford University Press, Oxford, 1996, pp. 343-349

[7] C.I. Christov An energy-consistent Galilean-invariant dispersive shallow-water model, Wave Motion, Volume 34 (2001), pp. 161-174

[8] A.C. Newell Solitons in Mathematics and Physics, SIAM, Philadelphia, 1985

[9] Ph. Rosenau Dynamics of nonlinear mass-spring chains near the continuum limit, Phys. Lett., Volume 118A (1987), pp. 222-227

[10] T.B. Benjamin; J.L. Bona; J.J. Mahony Model equation for long waves in nonlinear dispersive systems, Phil. Trans. Roy. Soc. London A, Volume 272 (1972), pp. 47-78

[11] L.A. Ostrovskii; A.M. Sutin Nonlinear waves in rods, J. Appl. Math. Mech., Volume 41 (1977), pp. 543-549 (English translation of P.M.M.)

[12] P.A. Clarkson; R.J. LeVeque; R. Saxton Solitary wave interactions in elastic rods, Stud. Appl. Math., Volume 75 (1986), pp. 95-122

[13] J.V. Boussinesq Théorie nouvelle des ondes lumineuses, J. Math. Pures Appl., Série 2, Volume 13 (1870), pp. 313-339

[14] G.A. Maugin Nonlinear Waves in Elastic Crystals, Oxford Univ. Press, Oxford, UK, 1999

[15] G.A. Nariboli Nonlinear longitudinal dispersive waves in elastic rods, J. Math. Phys. Sci., Volume 4 (1970), pp. 64-73

[16] L.A. Ostrovsky; A.I. Potapov Modulated Waves—Theory and applications, The Johns Hopkins Univ. Press, Baltimore, 1999

[17] A.M. Samsonov Strain Solitons in Solids and How to Construct Them, Chapman and Hall/CRC, Boca Raton, 2001

[18] G.A. Maugin; C.I. Christov Nonlinear duality between elastic waves and quasi-particles (C.I. Christov; A. Guran, eds.), Selected Topics in Nonlinear Wave Mechanics, Birkhäuser, Boston, 2002, pp. 117-160

[19] C.I. Christov Numerical investigation of the long-time evolution and interaction of localized waves (M.G. Velarde; C.I. Christov, eds.), Fluid Physics, World Scientific, Singapore, 1995, pp. 353-378

[20] C.I. Christov; M.G. Velarde Inelastic collisions of Boussinesq solitons, Int. J. Bifurcation Chaos, Volume 5 (1994), pp. 1095-1112

[21] M.J. Ablowitz; H. Segur Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981

[22] C.I. Christov; G.A. Maugin A numerical venture into the menagerie of coherent structures of generalized Boussinesq systems (M. Remoissenet; M. Peyrard, eds.), Coherent Structures in Physics and Biology, Springer, Berlin, 1991, pp. 206-215

[23] C.I. Christov; G.A. Maugin An implicit difference scheme for the long-time evolution of localized solutions of a generalized Boussinesq system, J. Comput. Phys., Volume 116 (1995), pp. 39-51

[24] C.I. Christov; G.A. Maugin Numerics of some generalized models of lattice dynamics (higher-order nonlinear and triple interactions) (J.L. Wegner; F. Norwood, eds.), Nonlinear Waves in Solids, AMS, vol. 137, ASME, New York, 1995, pp. 374-379

[25] C.I. Christov; G.A. Maugin; M.E. Velarde Well-posed Boussinesq paradigm with purely spatial higher-order derivatives, Phys. Rev. E, Volume 54 (1996), pp. 3621-3638

[26] H. Segur Who cares about integrability?, Physica D, Volume 51 (1991), pp. 343-359

[27] C.I. Christov; G.A. Maugin Long-time evolution of acoustic signals in nonlinear crystals (H. Hobaeck, ed.), Advances in Nonlinear Acoustics, World Scientific, Singapore, 1993, pp. 457-462

[28] A.M. Kosevich; M. Bogdan; G.A. Maugin Soliton complex dynamics in strongly dispersive systems, Wave Motion, Volume 34 (2001), pp. 1-26

[29] G.A. Maugin; S. Cadet Existence of solitary waves in martensitic alloys, Int. J. Engrg. Sci., Volume 29 (1991), pp. 243-258

[30] T. Kawahara Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, Volume 13 (1972), pp. 260-264

[31] T.T. Marinov; C.I. Christov; R.S. Marinova Novel numerical approach to solitary-wave solutions identification of Boussinesq and Korteweg–de Vries equation, Int. J. Bifurcation Chaos, Volume 15 (2005), pp. 557-565

[32] J.P. Boyd New directions in solitons and periodic waves: polycnoidal waves, imbricated solitons, weakly nonlocal solitary waves, and numerical boundary value algorithms (C.S. Yih, ed.), Advances in Applied Mechanics, vol. 27, Academic Press, New York, 1990, pp. 1-82

[33] G.A. Maugin On some generalizations of Boussinesq and KdV systems, Proc. Est. Acad. Sci. A, Volume 44 (1995), pp. 40-55 (special issue on the KdV equation)

[34] C.I. Christov; M.G. Velarde Dissipative solitons, Physica D, Volume 86 (1995), pp. 323-347

[35] C.I. Christov; M.G. Velarde Solitons and dissipation (M.G. Velarde; C.I. Christov, eds.), Fluid Physics, World Scientific, Singapore, 1995, pp. 472-506

[36] A.M. Samsonov Travelling wave solutions for nonlinear waves with dissipation, Appl. Anal., Volume 57 (1995), pp. 85-100

[37] A.V. Porubov; M.G. Velarde Dispersive-dissipative solitons in nonlinear solids, Wave Motion, Volume 31 (2000) no. 3, pp. 197-207

[38] A.V. Porubov Dissipative nonlinear strain waves in solids (C.I. Christov; A. Guran, eds.), Selected Topics in Nonlinear Wave Mechanics, Birkhäuser, Boston, 2002, pp. 223-260

[39] A.V. Porubov; M.G. Velarde Strain kinks in an elastic rod embedded in a viscoelastic medium, Wave Motion, Volume 35 (2001), pp. 189-204

[40] C.I. Christov; M.G. Velarde On localized solutions of an equation governing Bénard–Marangoni convection, Appl. Math. Modelling, Volume 17 (1993), pp. 311-320

[41] C.I. Christov; M.G. Velarde Evolution and interactions of solitary waves (solitons) in nonlinear dissipative systems, Physica Scripta, Volume T55 (1994), pp. 101-106

[42] C.I. Christov Dissipative quasi-particles: The generalized wave equation approach, Int. J. Bifurcation Chaos, Volume 12 (2002), pp. 2435-2444

[43] A.V. Porubov Amplification of Nonlinear Strain Waves in Solids, World Scientific, Singapore, 2003

[44] A.V. Porubov; G.A. Maugin; V.V. Mareev Localization of two-dimensional non-linear strain waves in a plate, Int. J. Non-Linear Mech., Volume 39 (2004), pp. 1359-1370

[45] A.V. Porubov; F. Pastrone; G.A. Maugin Selection of two-dimensional nonlinear strain waves in microstructure media, C. R. Mécanique (Acad. Sci. Paris), Volume 332 (2004), pp. 513-518

[46] A. Jeffrey; T. Kakutani Weak nonlinear dispersive waves: a discussion centered around the Korteweg–de Vries equation, SIAM Rev., Volume 14 (1972), pp. 582-643

[47] J. Choudhury; C.I. Christov 2D solitary waves of Boussinesq equation, Natchitoches, Oct. 2004 (APS Conference Proceedings), Volume 755 (2005), pp. 85-90

[48] M.A. Christou; C.I. Christov Fourier–Galerkin method for 2D solitons of Boussinesq equation, Mathematics and Computers in Simulation, Volume 74 (2007), pp. 82-92

[49] A. Salupere; G.A. Maugin; Jü. Engelbrecht Solitonic structures in KdV-based higher order systems, Wave Motion, Volume 34 (2001), pp. 51-61

[50] A.V. Porubov; V.V. Gursky; V.V. Krzhizhanovskaya; G.A. Maugin On some localized waves described by the extended KdV equation, C. R. Mécanique (Acad. Sci. Paris), Volume 333 (2005), pp. 528-533

[51] N. Flytzanis; St. Pnevmatikos; M. Remoissenet Kink, breather and asymmetric envelope or dark solitons in nonlinear chains-I-monoatomic chain, J. Phys. C Solid State Phys., Volume 18 (1986), pp. 4603-4629

[52] G.B. Whitham Linear and Nonlinear Waves, Wiley–Interscience, New York, 1974

[53] G.A. Maugin Nonlinear kinematic-wave mechanics of elastic solids, Wave Motion, Volume 44 (2007), pp. 472-481

[54] K. Bataille; F. Lund Nonlinear waves in elastic media, Physica D, Volume 6 (1982), pp. 95-105

[55] G.A. Maugin Theory of nonlinear surface waves and solitons (C.G. Lai; K. Wilmanski, eds.), Surface Waves in Geomechanics, Springer, Wien, 2005, pp. 325-371

[56] G.A. Maugin Application of an energy-momentum tensor in nonlinear elastodynamics: pseudomomentum and Eshelby stress in solitonic elastic systems, J. Mech. Phys. Solids, Volume 29 (1992), pp. 1543-1558

[57] Yu.S. Kivshar; B.A. Malomed Dynamics of solitons in nearly integrable systems, Rev. Modern Phys., Volume 61 (1989), pp. 763-915

[58] G.A. Maugin; C.I. Christov Nonlinear duality between elastic waves and quasi-particles in microstructured solids, Tallinn, 1996 (Proc. Est. Acad. Sci. A), Volume 46 (1997), pp. 78-84

[59] E. Meletlidou; J. Pouget; G.A. Maugin; E.C. Aifantis Invariant relations in Boussinesq type equations, Chaos Solitons Fractals (J. Phys. UK), Volume 22 (2004) no. 3, pp. 613-625

[60] C.I. Christov A complete orthonormal sequence of functions in $L^{2} (- \infty, \infty)$ space, SIAM J. Appl. Math., Volume 42 (1982), pp. 1337-1344

[61] V. Varlamov, Two-dimensional Boussinesq equation in a disc and anisotropic Sobolev spaces, C. R. Mecanique (2007), this issue; | DOI

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