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Integrable Nonlinear Wave Equations and Possible Connections to Tsunami Dynamics

  • M. Lakshmanan

Summary

In this article we present a brief overview of the nature of localized solitary wave structures/solutions underlying integrable nonlinear dispersive wave equations with specific reference to shallow water wave propagation and explore their possible connections to tsunami waves. In particular, we will discuss the derivation of Korteweg-de Vries family of soliton equations in unidirectional wave propagation in shallow waters and their integrability properties and the nature of soliton collisions.

Keywords

Solitary Wave Tsunami Wave Solitary Wave Solution Shallow Water Wave Cnoidal Wave 
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References

  1. [1]
    Lay T etal (2005) The Great Sumatra-Andaman earthquake of 26 December 2004. Science 308:1127–1133CrossRefGoogle Scholar
  2. [2]
    Ram Mohan V etal (2006) Impact of South Asian Tsunami on 26 December 2004 on south east coast of India-A field report (preprint)Google Scholar
  3. [3]
    Dudley WC and Min L (1988) Tsunami!. Honolulu, Hawaii: Univeristy of Hawaii pressGoogle Scholar
  4. [4]
    Okada Y (1985) Surface deformation due to sheer and tensile fault in a half space. Bull Seism Soc Am 82:1135–54Google Scholar
  5. [5]
    Gower J (2005) Jason 1 detects the December 2004 tsunami. EOS 86:37–38CrossRefGoogle Scholar
  6. [6]
    Constantin A and Johnson RS (2006) Modelling tsunami. J Phys A: Math Gen 39:L215–L217CrossRefGoogle Scholar
  7. [7]
    See for example, Bullough RK (1988) The wave par excellence: the solitary progressive great wave of equilibrium of the fluid-an early history of solitary wave. In: Lakshmanan M (Ed.) Solitons: Introduction and applications. Springer, BerlinGoogle Scholar
  8. [8]
    Kortweg DJ and de Vries D (1895) On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil Mag 39:422–443Google Scholar
  9. [9]
    Lakshmanan M and Rajasekar S (2003) Nonlinear Dynamics: Integrability, Chaos and Patterns. Springer, BerlinGoogle Scholar
  10. [10]
    Ablowitz MJ and Clarkson PA (1991) Solitons, Nonlinear evolution equations and inverse scattering. Cambridge University Press, CambridgeGoogle Scholar
  11. [11]
    Zabusky NJ and Kruskal MD (1965) Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15:240–43CrossRefGoogle Scholar
  12. [12]
    Gardner CS, Greene JM, Kruskal MD and Miura RM (1967) Method for solving the Kortweg-de Vries equation, Phys Rev Lett 19:1095–97CrossRefGoogle Scholar
  13. [13]
    Zakharov VE and Faddeev LD (1971) The Kortweg-de Vries equation is a fully integrable Hamiltonian system. Funct Anal Appl 5:280–87CrossRefGoogle Scholar
  14. [14]
    Hirota R (1971) Exact solution of the Kortweg-de Vries equation for multiple collisions of solitons. Phys Rev Lett 27:1192–94CrossRefGoogle Scholar
  15. [15]
    Camassa R and Holm D (1993) An integrable shallow water equation with peaked solitons. Phys Rev Lett 71:1661–64CrossRefGoogle Scholar
  16. [16]
    Calogero F and Degasperis A (1982) Spectral transform and solitons. North-Holland, AmsterdamGoogle Scholar
  17. [17]
    Caputo JG and Stepanyants YA (2003) Bore formation, evolution and disintegration into solitons in shallow inhomogeneous channels. Nonlinear Processes in Geophysics 10:407–24Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • M. Lakshmanan
    • 1
  1. 1.Centre for Nonlinear Dynamics, School of PhysicsBharathidasan UniversityTiruchirapalli

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