Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Solitons, Tsunamis and Oceanographical Applications of

  • M. Lakshmanan
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_509-3

Definition of the Subject

Surface and internal gravity waves arising in various oceanographic conditions are natural sources where one can identify/observe the generation, formation, and propagation of solitary waves and solitons. Unlike the standard progressive waves of linear dispersive type, solitary waves are localized structures with long wavelengths and finite energies and propagate without change of speed or form and are patently nonlinear entities. The earliest scientifically recorded observation of a solitary wave was made by John Scott Russel in August 1834 in the Union Canal connecting the Scottish cities of Glasgow and Edinburgh. The theoretical formulation of the underlying phenomenon was provided by Korteweg and de Vries in 1895 who deduced the now-famous Korteweg-de Vries (KdV) equation admitting solitary wave solutions. With the insightful numerical and analytical investigations of Martin Kruskal and coworkers in the 1960s, the KdV solitary waves have been shown to...

Keywords

Solitary Wave Rossby Wave Water Wave Tsunami Wave Solitary Wave Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Bibliography

Primary Literature

  1. Ablowitz MJ, Clarkson PA (1991) Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  2. Ablowitz MJ, Segur H (1981) Solitons and inverse scattering transform. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  3. Alpers W, Wang-Chen H, Cook L (1997) Observation of internal waves in the Andaman sea by ERS SAR. IEEE Int 4:1518–1520Google Scholar
  4. Apel JR, Holbroook JR (1980) The Sulu sea internal soliton experiment, 1. Background and overview. Eos Trans AGU 61:1009Google Scholar
  5. Banerjee P, Politz FF, Burgman R (2005) The size and duration of the Sumatra-Andaman earthquake from far-field static offsets. Science 308:1769–1772ADSCrossRefGoogle Scholar
  6. Benjamin TB (1967) Internal waves of permanent form in fluids of great depth. J Fluid Mech 29:559–592ADSCrossRefzbMATHGoogle Scholar
  7. Benjamin TB, Bona JL, Mahoney JJ (1972) Model equations for long waves in nonlinear dispersive systems. Philos Trans A R Soc 272:47–78ADSCrossRefzbMATHGoogle Scholar
  8. Benney DJ (1966) Long nonlinear waves in fluid flows. Stud Appl Math 45:52–63zbMATHMathSciNetGoogle Scholar
  9. Benney DJ, Roskes G (1969) Wave instabilities. Stud Appl Math 48:377–385zbMATHGoogle Scholar
  10. Boyd JP (1980) Equatorial solitary waves. Part I: Rossby solitons. J Phys Oceanogr 10:1699–1717ADSCrossRefGoogle Scholar
  11. Bullough RK (1988) The wave par excellence. The solitary progressive great wave of equilibrium of fluid: an early history of the solitary wave. In: Lakshmanan M (ed) Solitons: introduction and applications. Springer, BerlinGoogle Scholar
  12. Camassa R, Holm D (1992) An integrable shallow water equation with peaked solitons. Phys Rev Lett 71:1661–1664ADSCrossRefMathSciNetGoogle Scholar
  13. Caputo JG, Stepanyants YA (2003) Bore formation and disintegration into solitons in shallow inhomogeneous channels. Nonlinear Process Geophys 10:407–424ADSCrossRefGoogle Scholar
  14. Caputo JG, Stepanyants YA (2007) Tsunami surge in a river: a hydraulic jump in an inhomogeneous channel. In: Kundu A (ed) Tsunami and nonlinear waves. Springer, Berlin, pp 97–112CrossRefGoogle Scholar
  15. Carrier GF, Wu TT, Yeh H (2003) Tsunami runup and drawdown on a plane beach. J Fluid Mech 475:79–99ADSzbMATHMathSciNetGoogle Scholar
  16. Constantin A, Johnson RS (2006) Modelling tsunamis. J Geophys Res A39:L215–L217MathSciNetGoogle Scholar
  17. Davey A, Stewartson K (1974) On three dimensional packets of surface waves. Proc R Soc Lond A 338:101–110ADSCrossRefzbMATHMathSciNetGoogle Scholar
  18. Dias F, Dutykh D (2007) Dynamics of tsunami waves. In: Ibrahimbegovic A, Kozar I (eds) Extreme man-made and natural hazards in dynamics of structures, NATO security through science series. Springer, Berlin, pp 201–224CrossRefGoogle Scholar
  19. Dudley WC, Miu L (1988) Tsunami! University of Hawaii Press, HonoluluGoogle Scholar
  20. Dutykh D, Dias F (2007) Water waves generated by a moving bottom. In: Kundu A (ed) Tsunami and nonlinear waves. Springer, Berlin, pp 65–94CrossRefGoogle Scholar
  21. Fokas AS, Santini PM (1990) Dromions and a boundary value problem for the Davey-Stewartson I equation. Physica D 44:99–130ADSCrossRefMathSciNetGoogle Scholar
  22. Gardner CS, Greene JM, Kruskal MD, Miura RM (1967) Method for solving the Korteweg-de Vries equation. Phys Rev Lett 19:1095–1097ADSCrossRefGoogle Scholar
  23. Hasimoto H, Ono H (1972) Nonlinear modulation of gravity waves. J Phys Soc Jpn 33:805–811ADSCrossRefGoogle Scholar
  24. Helal MA, Molines JM (1981) Nonlinear internal waves in shallow water. A theoretical and experimental study. Tellus 33:488–504ADSCrossRefMathSciNetGoogle Scholar
  25. Hyder P, Jeans DRG, Cauqull E, Nerzic R (2005) Observations and predictability of internal solitons in the northern Andaman sea. Appl Ocean Res 27:1–11CrossRefGoogle Scholar
  26. Joseph RI (1977) Solitary waves in a finite depth fluid. J Geophys Res A10:L225–L227Google Scholar
  27. Koch SE, Pagowski M, Wilson JW, Fabry F, Flamant C, Feltz W, Schwemmer G, Geerts B (2005) The structure and dynamics of atmospheric bores and solitons as determined from remote sensing and modelling experiments during IHOP. In: AMS 32nd conference on radar meteorology, report JP6J.4Google Scholar
  28. Korteweg DJ, de Vries G (1895) On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos Mag 39:422–443CrossRefzbMATHGoogle Scholar
  29. Kundu A (ed) (2007) Tsunami and nonlinear waves. Springer, BerlinzbMATHGoogle Scholar
  30. Kundu PK, Cohen IM (2002) Fluid mechanics, 2nd edn. Academic, San DiegoGoogle Scholar
  31. Lakshmanan M (2007) Integrable nonlinear wave equations and possible connections to tsunami dynamics. In: Kundu A (ed) Tsunami and nonlinear waves. Springer, Berlin, pp 31–49CrossRefGoogle Scholar
  32. Lakshmanan M, Rajasekar S (2003) Nonlinear dynamics: integrability and chaos. Springer, BerlinCrossRefGoogle Scholar
  33. Lee S, Held I (1993) Baroclinic wave packets in models and observations. J Atmos Sci 50:1413–1428ADSCrossRefGoogle Scholar
  34. Longuet-Higgins MS (1993) Capillary gravity waves of solitary type and envelope solitons in deep water. J Fluid Mech 252:703–711ADSCrossRefzbMATHMathSciNetGoogle Scholar
  35. Maxworthy T, Redekopp LG (1976) Theory of the Great Red Spot and other observed features of the Jovian atmosphere. Icarus 29:261–271ADSCrossRefGoogle Scholar
  36. Okada Y (1992) Internal deformation due to shear and tensile faults in a half space. Bull Seism Soc Am 82:1018–1040Google Scholar
  37. Osborne AR, Burch TL (1980) Internal solitons in the Andaman sea. Science 258:451–460ADSCrossRefGoogle Scholar
  38. Ostrovsky LA, Stepanyants YA (2005) Internal solitons in laboratory experiments: comparison with theoretical models. Chaos 15(1–28):037111ADSCrossRefMathSciNetGoogle Scholar
  39. Philips OM (1974) Nonlinear dispersive waves. Annu Rev Fluid Mech 6:93–110ADSCrossRefGoogle Scholar
  40. Redekopp L (1977) On the theory of solitary Rossby waves. J Fluid Mech 82:725–745ADSCrossRefzbMATHMathSciNetGoogle Scholar
  41. Rossby GG (1939) Relation between variations in the intensity of the zonal circulation of the atmosphere. J Mar Res 2:38–55CrossRefGoogle Scholar
  42. Russel JS (1844) Reports on waves. 14th meeting of the British Association for advancement of science. John Murray, London, pp 311–390Google Scholar
  43. Scott AC (1999) Nonlinear science: emergence and dynamics of coherent structures. Oxford University Press, New YorkzbMATHGoogle Scholar
  44. Segur H (2007) Waves in shallow water, with emphasis on the tsunami of (2004). In: Kundu A (ed) Tsunami and nonlinear waves. Springer, Berlin, pp 3–29CrossRefGoogle Scholar
  45. Susanto RD, Zheng Q, Xiao-Hai Y (1998) Complex singular value decomposition analysis of equatorial waves in the Pacific observed by TOPEX/Poseidon altimeter. J Atmos Oceanic Tech 15:764–774CrossRefGoogle Scholar
  46. Tan B (1996) Collision interactions of envelope Rossby solitons in a baratropic atmosphere. J Atmos Sci 53:1604–1616ADSCrossRefGoogle Scholar
  47. Tsuji T, Yanuma T, Murata I, Fujiwara C (1991) Tsunami ascending in rivers as an undular bore. Nat Hazard 4:257–266CrossRefGoogle Scholar
  48. Zabusky NJ, Kruskal MD (1965) Interactions of solitons in a collisionless plasma and recurrence of initial states. Phys Rev Lett 15:240–243ADSCrossRefzbMATHGoogle Scholar
  49. Zakharov VE (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. J Appl Mech Tech Phys 2:190–194ADSGoogle Scholar

Books and Reviews

  1. Bourgault D, Richards C (2007) A laboratory experiment on internal solitary waves. Am J Phys 75:666–670ADSCrossRefGoogle Scholar
  2. Dauxois T, Peyrard M (2006) Physics of solitons. Cambridge University Press, CambridgezbMATHGoogle Scholar
  3. Drazin PG, Johnson RS (1989) Solitons: an introduction. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  4. Hammack JL (1973) A note on tsunamis: their generation and propagation in an ocean of uniform depth. J Fluid Mech 60:769–800ADSCrossRefzbMATHGoogle Scholar
  5. Helfrich KR, Melville WK (2006) Long nonlinear internal waves. Annu Rev Fluid Mech 38:395–425ADSCrossRefMathSciNetGoogle Scholar
  6. Johnson RS (1997) An introduction to the mathematical theory of water waves. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  7. Lamb H (1932) Hydrodynamics. Dover, New YorkzbMATHGoogle Scholar
  8. Mei CC (1983) The applied dynamics of ocean surface waves. Wiley, New YorkzbMATHGoogle Scholar
  9. Miles JW (1980) Solitary waves. Annu Rev Fluid Mech 12:11–43ADSCrossRefGoogle Scholar
  10. Scott AC (ed) (2005) Encyclopedia of nonlinear science. Routledge, New YorkzbMATHGoogle Scholar
  11. Scott AC et al (1973) The soliton: a new concept in applied science. Proc IEEE 61:1443–1483ADSCrossRefMathSciNetGoogle Scholar
  12. Stoker JJ (1957) Water waves. Interscience, New YorkzbMATHGoogle Scholar
  13. Sulem C, Sulem P (1999) The nonlinear Schrödinger equation. Springer, BerlinzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Center for Nonlinear DynamicsBharathidasan UniversityTiruchirapalliIndia