Definition of the Subject
The interest in nonlinear physics has grown significantly over the last 50 years. Although numerous nonlinear processes had been previously identified, the mathematic tools of nonlinear physics had not yet been developed. The available tools were linear, and nonlinearities were avoided or treated as perturbations of linear theories. The solitary water wave, experimentally discovered in 1834 by John Scott Russell, led to numerous discussions. This hump-shape localized wave that propagates along one space direction with undeformed shape has spectacular stability properties. John Scott Russell carried out many experiments to obtain the properties of this wave. The theories which were based on linear approaches concluded that this kind of wave could not exist. The controversy was resolved by J. Boussinesq (1871) and by Lord Rayleigh (1876) who showed that if dissipation is neglected, the increase in local wave velocity associated with finite amplitude is balanced...
Similar content being viewed by others
Bibliography
Primary Literature
Ablowitz MJ, Clarkson PA (1991) Solitons, nonlinear evolutions equations and inverse scattering. Cambridge University Press, Cambridge
Airy GB (1845) Tides and waves. Encycl Metropolitana 5:291–396
Bazin H (1865) Recherches expérimentales relatives aux remous et à la propagation des ondes. In: Darcy H, Bazin H (eds) Recherches hydrauliques. Imprimerie impériale, Paris
Benjamin TB, Feir JE (1967) The disintegration of wavetrains on deep water. Part 1 Theory. J Fluid Mech 27:417–430
Boussinesq J (1871) Théorie de l'intumescence liquide appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire. CR Acad Sci 72:755–759
Boussinesq J (1877) Essai sur la théorie des eaux courantes. MSE 23:1–680
Brandt P, Rubino A, Alpers W, Backhaus JO (1997) Internal waves in the Strait of Messina studied by a numerical model and synthetic aperture radar images from the ERS 1/2 satellites. J Phys Oceanogr 27:648–663
Bullough RK (1988) The wave par excellence, the solitary progressive great wave of equilibrium of the fluid: an early history of the solitary wave. In: Lakshmanan M (ed) Solitons: introduction and applications, Springer series nonlinear dynamics. Springer, New York, pp 150–281
Canuto C, Hussaini MY, Quarteroni A, Zang TA (1988) Spectral methods in fluid dynamics. Springer, Berlin
Su CH, Gardner CS (1969) Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation. J Math Phys 10:536–539
Chanson H (2005) Le tsunami du 26 décembre 2004: un phénomène hydraulique d'ampleur internationale. Premiers constats. Houille Blanche 2:25–32
Darrigol O (2003) The spirited horse, the engineer, and the mathematician: water waves in nineteenth- century hydrodynamics. Arch Hist Exact Sci 58:21–95
Dias F, Dutykh D (2007) Dynamics of tsunami waves. Extreme man-made and natural hazards in dynamics of structures. In: Ibrahimbegovic A, Kozar I (eds) Proceedings of NATO advanced research workshop on extreme man-made and natural hazards in dynamics of structures. Springer, Opatija
Donnelly C, Chanson H (2002) Environmental impact of a tidal bore on tropical rivers. In: Proceedings of 5th international river management symposium, Brisbane
Edler J, Hamm P (2002) Self-trapping of the amide I band in a peptide model crystal. J Chem Phys 117:2415–2424
Emerson GS (1977) John Scott Russell: a great victorian engineer and naval architect. John Murray, London
Ezersky AB, Polukhina OE, Brossard J, Marin F, Mutabazi I (2006) Spatiotemporal properties of solitons excited on the surface of shallow water in a hydrodynamic resonator. Phys Fluids 18:067104
Fermi E, Pasta J, Ulam S (1955) Studies of nonlinear problems. Los Alamos report, LA-1940. published later. In: Segré E (ed) Collected papers of enrico fermi. University of Chicago Press. 1965
Fochesato C, Grilli S, Dias F (2007) Numerical modelling of extreme rogue waves generated by directional energy focusing. Wave Motion 44:395–416
Ford JJ (1961) Equipartition of energy for nonlinear systems. J Math Phys 2:387–393
Fornberg B, Whitham GB (1978) A numerical and theoretical study of certain nonlinear wave phenomena. Philos Trans R Soc Lond 289:373–404
Frenkel J, Kontorova T (1939) On the theory of plastic deformation and twinning. J Phys 1:137–149
Gardner CS, Morikawa GK (1960) Similarity in the asymptotic behaviour of collision-free hydromagnetic waves and water waves. Technical Report NYO-9082, Courant Institute of Mathematical Sciences. New York University, New York
Gardner CS, Green JM, Kruskal MD, Miura RM (1967) Method for solving the Korteweg-de Vries equation. Phys Rev Lett 19:1095–1097
Gerkema T, Zimmerman JTF (1994) Generation of nonlinear internal tides and solitary waves. J Phys Oceanogr 25:1081–1094
Goring DG (1978) Tsunamis – the propagation of long waves onto a shelf. PhD thesis, California Inst Techn, Pasadena
Greig IS, Morris JL (1976) A hopscotch method for the KdV equation. J Comput Phys 20:64–80
Guizien K, Barthélemy E (2002) Accuracy of solitary wave generation by a piston wave maker. J Hydraul Res 40(3):321–331
Hammack JL, Segur H (1974) The Korteweg-de Vries equation and water waves. Part 2. Comparisons with experiments. J Fluid Mech 65:289–314
Hao R, Li L, Li ZH, Xue W, Zhou GS (2004) A new approach to exact soliton solutions and soliton interaction for the nonlinear Schrödinger equation with variable coefficients. Opt Commun 236:79–86
Hasegawa A, Tappert F (1973) Transmission of stationary nonlinear optical pulses in dispersive dielectric fiber: II. Normal dispersion. Appl Phys Lett 23:171–172
Hashizume Y (1985) Nonlinear pressure waves in a fluid-filled elastic tube. J Phys Soc Japan 54:3305–3312
Hashizume Y (1988) Nonlinear pressure wave propagation in arteries. J Phys Soc Japan 57:4160–4168
Heeger AJ, Kivelson S, Schrieffer JR, WP S (1988) Solitons in conducting polymers. Rev Mod Phys 60:781–850
Helal MA (2001) Chebyshev spectral method for solving KdV equation with hydrodynamical application. Chaos Solit Fractals 12:943–950
Helal MA (2002) Review: soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics. Chaos Solit Fractals 13:1917–1929
Helal MA, El-Eissa HN (1996) Shallow water waves and KdV equation (oceanographic application). PUMA 7(3–4):263–282
Hirota R (1971) Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys Rev Lett 27:1192–1194
Jackson EA (1963) Nonlinear coupled oscillators. I. Perturbation theory: ergodic problems. J Math Phys 4:551–558
Jaworski M, Zagrodzinski J (1995) Position and position-like solution of KdV and Sine-Gordon equations. Chaos Solit Fractals 5(12):2229–2233
Kharif C, Pelinovsky E (2003) Physical mechanisms of the rogue wave phenomenon. Eur J Mech B/Fluids 22:603–634
Korteweg DJ, De Vries G (1895) On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves. Philos Mag 39(5):442–443
Kruglov VI, Peacok AC, Harvey JD (2003) Exact self-similar solutions of the generalized nonlinear schrödinger equation with distributed coefficients. Phys Rev Lett 90(11):113902. http://prola.aps.org/abstract/PRL/v90/i11/e113902
Lakshmanan M (1997) Nonlinear physics: integrability, chaos and beyond. J Frankl Inst 334B(5/6):909–969
Lakshmanan M, Sahadevan R (1993) Painlevé analysis, Lie symmetries and integrability of coupled nonlinear oscillators of polynomial type. Phys Rep 224:1–93
Lamb H (1879) Treatise on the motion of fluids. In: Hydrodynamics, 6th edn. Cambridge University Press, Cambridge. 1952
Lamb H (1971) Analytical descriptions of ultrashort optical pulse propagation in a resonant medium. Rev Mod Phys 43:99–124
Liu PL-F, Synolakis CE, Yeh HH (1991) Report on the international workshop on long-wave run-up. J Fluid Mech 229:675–688
Lo EYM, Shao S (2002) Simulation of near-shore solitary wave mechanisms by an incompressible SPH method. Appl Ocean Res 24:275–286
Lynch DK (1982) Tidal bores. Sci Am 247:134–143
Malfliet W (1992) Solitary wave solutions of nonlinear wave equations. Am J Phys 60(7):650–654
Marin F, Abcha N, Brossard J, Ezersky AB (2005) Laboratory study of sand bedforms induced by solitary waves in shallow water. J Geophys Res:110(F4):F04S17
Martnez Alonso L, Olmedilla Morino E (1995) Algebraic geometry and soliton dynamics. Chaos Solit Fractals 5(12):2213–2227
Maxworthy T (1976) Experiments on collision between solitary waves. J Fluid Mech 76:177–185
Mei CC, Li Y (2004) Evolution of solitons over a randomly rough seabed. Phys Rev E 70:016302
Michallet H, Barthélemy E (1998) Experimental study of interfacial solitary waves. J Fluid Mech 366:159–177
Mireska HJ, Steiner M (1991) Solitary excitations in one-dimensional magnets. Adv Phys 40:196–356
Mollenauer LF, Stolen RH, Gordon JP (1980) Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys Rev Lett 45:1095–1098
Newell AC (1985) Solitons in mathematics and physics. SIAM, Philadelphia
Olver PJ (1986) Applications of lie groups to differential equations, Graduate texts in mathematics, vol 107. Springer, Berlin
Ostrovsky LA, Stepanyants YA (1989) Do internal solitons exist in the ocean? Rev Geophys 27:293–310
Paquerot JF, Remoissenet M (1994) Dynamics of nonlinear blood pressure waves in large arteries. Phys Lett A 194:77–82
Peyrard M, Dauxois T (2004) Physique des solitons. EDP Sciences/CNRS Editions, Les Ulis/Paris
Rayleigh L (1876) On waves. Philos Mag 5(1):257–279
Remoissenet M (1999) Waves called solitons – concepts and experiments. Springer, Berlin
Rodwell MJ, Allen ST, RY Y, Case MG, Bhattacharya U, Reddy M, Carman E, Kamegawa M, Konishi Y, Pusl J, Pullela R (1994) Active and nonlinear wave propagation devices in ultrafast electronics and optoelectronics. Proc IEEE 82:1037–1058
Rogers C, Shadwick WF (1982) Bäcklund transformations and applications. Academic, New York
Rottman JW, Grimshaw R (2001) Atmospheric internal solitary waves. In: Environmental stratified flows. Kluwer, Boston, pp 61–88
Russell JS (1837) Report on the committee on waves. In: Murray J (ed) Bristol, Brit Ass Rep, London, pp 417–496
Russell JS (1839) Experimental researches into the laws of certain hydrodynamical phenomena that accompany the motion of floating bodies, and have not previously been reduced into conformity with the known laws of the resistance of fluids. Trans Royal Soc Edinb 14:47–109
Russell JS (1844) Report on waves. In: Murray J (ed) Brit Ass Rep Adv Sci 14, London, pp 311–390
Sanz-Serna JM, Christie I (1981) Petrov-Galerkin method for nonlinear dispersive waves. J Comput Phys 39:94–102
Stokes GS (1847) On the theory of oscillatory waves. Trans Cambridge Phil Soc 8:441–473
Taha TR, Ablowitz MJ (1984) Analytical and numerical aspects of certain nonlinear evolution equations, (III) numerical, KdV equation. J Comput Phys 55:231–253
Taniuti T, Wei CC (1968) J Phys Soc Jpn 24:941–946
Tappert F (1974) Numerical solution of the KdV equation and its generalisation by split-step Fourier method. Lec Appl Math Am Math Soc 15:215–216
Ustinov AV (1998) Solitons in josephson junctions. Phys D 123:315–329
Weidman PD, Maxworthy T (1978) Experiments on strong interaction between solitary waves. J Fluid Mech 85:417–431
Yomosa S (1987) Solitary waves in large blood vessels. J Phys Soc Japan 56:506–520
Yuen HC, Lake BM (1975) Nonlinear deep water waves: theory and experiments. Phys Fluids 18:956–960
Zabusky NJ, Kruskal MD (1965) Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15:240–243
Zakharov VE, Shabat AB (1972) Exact theory of two-dimensional self-focusing and one dimensional self-modulation of waves in nonlinear media. Sov Phys JETP 34:62–69
Books and Reviews
Agrawal GP (2001) Nonlinear fiber optics. Academic Press, Elsevier
Akmediev NN, Ankiewicz A (1997) Solitons, nonlinear pulses and beams. Chapman and Hall, London
Braun OM, Kivshar YS (2004) The frenkel-kontorova model. concepts, methods and applications. Springer, Berlin
Bullough RK, Caudrey P (1980) Solitons. Springer, Heidelberg
Cooker MJ, Weidman PD, Bale DS (1997) Reflection of high amplitude wave at a vertical wall. J Fluid Mech 342:141–158
Davydov AS (1985) Solitons in molecular systems. Reidel, Dordrecht
Dodd RK, Eilbeck JC, Gibbon JD, Morris HC (1982) Solitons and nonlinear wave equations. Academic Press, London
Drazin PG, Johnson RS (1993) Solitons: an introduction. Cambridge University Press, Cambridge
Eilenberger G (1981) Solitons: mathematical methods for physicists. Springer, Berlin
Hasegawa A (1989) Optical solitons in fibers. Springer, Heidelberg
Infeld E, Rowlands G (2000) Nonlinear waves, solitons and chaos. Cambridge University Press, Cambridge
Lamb GL (1980) Elements of soliton theory. Wiley, New York
Toda M (1978) Theory of nonlinear lattices. Springer, Berlin
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer Science+Business Media LLC
About this entry
Cite this entry
Marin, F. (2017). Solitons: Historical and Physical Introduction. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_506-2
Download citation
DOI: https://doi.org/10.1007/978-3-642-27737-5_506-2
Received:
Accepted:
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-27737-5
Online ISBN: 978-3-642-27737-5
eBook Packages: Springer Reference Physics and AstronomyReference Module Physical and Materials ScienceReference Module Chemistry, Materials and Physics