Abstract
A solitary wave that, eventually, becomes a soliton represents a unique and very attractive object in modern nonlinear science of spatially extended systems. The term “soliton” was coined by N.J. Zabusky and M.D. Kruskal [2.30] to characterize solitary waves in nonlinear systems whose properties (energy etc.) are mostly localized in a bounded region of space at any instant of time such that upon collision they act like particles (e.g. electron, proton, etc.) and, in the case they studied, elastically, crossing each other or interchanging places with no significant change of form.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bazin, H., “Recherches experimentales sur la propagation des ondes”, Mém. présentés par divers savants à l’ Acad. Sci. Inst. France (Paris) 19 (1865) 495–644.
Bishop, A. R., Krumhansl, J. A. and Trullinger, S. E., “Solitons in condensed matter: A paradigm”, Physica D 1 (1980) 1–44.
Bouasse, H., Houle, Rides, Seiches et Marées (Delagrave, Paris, 1924), pp. 291–292.
Boussinesq, J. V., “Théorie de l’intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire”, C. R. Hebd. Séances Acad. Sci. (Paris) 72 (1871) 755–759.
Boussinesq, J. V., “Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond”, J. Math. Pures Appl. 17 (1872) 55–108.
Boussinesq, J. V., “Essai sur la théorie des eaux courantes”, Mém. présentés par divers savants à l’Acad. Sci. Inst. France (Paris) 23 (1877) 1–680. Additions et éclaircissements au mémoire intitulé (supra), Mém. présentés par divers savants à l’Acad. Sci. Inst. France (Paris) ibidem 24[2] (1878) 1–64.
Christov, C. I. and Velarde, M. G., “Dissipative solitons”, Physica D 86 (1995) 323–347.
Christov, C. I., Maugin, G. A. and Velarde, M. G., “Well-posed Boussinesq paradigm with purely spatial higher-order derivatives”, Phys. Rev. E 54 (1996) 3621–3638.
Dodd, R. K., Eilbeck, J. C., Gibbon, J. D. and Morris, H. C., Solitons and Nonlinear Wave Equations (Academic Press, Inc., 1984).
Drazin, P. G. and Johnson, R. S., Solitons: An Introduction (University Press, Cambridge, 1989).
Garazo, A. N. and Velarde, M. G., “Dissipative Korteweg-de Vries description of Marangoni-Bénard oscillatory convection”, Phys. Fluids A 3 (1991) 2295–2300.
Gardner, P. L., Greene, J. M., Kruskal, M. D. and Miura, R. M., “Method for solving Korteweg-de Vries equation”, Phys. Rev. Lett. 19 (1967) 1095–1097.
Kliakhandler, I. L., Porubov, A. V. and Velarde, M. G., “Localized finiteamplitude disiturbances and selection of solitary waves”, Phys. Rev. E 62 (2000) 4959–4962.
Korteweg, D. J. and de Vries G., “On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves”, Phil. Mag. 39[5] (1895) 422–443.
Krehl, P. and van der Geest, M., “The discovery of the Mach reflection effect and its demonstration in an auditorium”, Shock Waves 1 (1991) 3–15.
Linde, H., Chu, X.-L. and Velarde, M. G., “Oblique and head-on collisions of solitary waves in Marangoni-Bénard convection”, Phys. Fluids A 5 (1993) 1068–1070.
Linde, H., Chu, X.-L., Velarde, M. G. and Waldhelm, W., “Wall reflections of solitary waves in Marangoni-Bénard convection”, Phys. Fluids A 5 (1993) 3162–3166.
Mach, E. and Wosyka, J., “Über einige mechanische Wirkungen des elektrischen Funkens”, Sitzungsber. Akad. Wiss. Wien 72[II] (1875) 44–52.
Nekorkin, V. I. and Velarde, M. G., “Solitary waves, soliton bound states and chaos in a dissipative Korteweg-de Vries equation”, Int J. Bifurcation Chaos 4 (1994) 1135–1146.
Nepomnyashchy, A. A., Velarde, M. G. and Colinet, P., Interfacial Phenomena and Convection (Chapman & Hall/CRC, New York, 2002).
Newell, A. C., Solitons in Mathematics and Physics (SIAM, Philadelphia, 1985).
Rayleigh, Lord, “On waves”, Phil. Mag. 1 (1876) 257–279.
Rednikov, A. Y., Velarde, M. G., Ryazantsev, Yu. S., Nepomnyashchy, A. A. and Kurdyumov, V. N., “Cnoidal Wave Trains ans Solitary Waves in a Dissipation-Modified Korteweg-de Vries Equation”, Acta Appl. Math. 39 (1995) 457–475.
Remoissenet, M., Waves Called Solitons. Concepts and Experiments, 3rd Edition (Springer-Verlag, Berlin, 2001).
Russell, J.S., “Report on waves”, Rep. 14th Meet. British Ass. Adv. Sci., York, 311–390, (J. Murray, London, 1844).
Russell, J. S., The Wave of Translation in the Oceans of Water, Air and Ether, with Appendix (Trübner, London, 1885).
Ursell, F., “The long-wave paradox in the theory of gravity waves”, Proc. Cambridge Phil. Soc. 49 (1953) 685–694.
Velarde, M. G., Nekorkin, V. I. and Maksimov, A. G., “Further results on the evolution of solitary waves and their bound states of a dissipative Korteweg-de Vries equation”, Int. J. Bifurcation Chaos 5 (1995) 831–839.
Velarde, M. G., Nepomnyashchy, A. A. and Hennenberg, M., “Onset of oscillatory interfacial instability and wave motions in Bénard layers”, Adv. Appl. Mech. 37 (2000) 167–238.
Zabusky, N. and Kruskal, M. D., “Interaction of’ solitons’ in a collisionless plasma and the recurrence of initial states”, Phys. Rev. Lett. 15 (1965) 57–62.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Nekorkin, V.I., Velarde, M.G. (2002). Solitary Waves, Bound Soliton States and Chaotic Soliton Trains in a Dissipative Boussinesq-Korteweg-de Vries Equation. In: Synergetic Phenomena in Active Lattices. Springer Series in Synergetics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56053-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-56053-8_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-62725-5
Online ISBN: 978-3-642-56053-8
eBook Packages: Springer Book Archive