Nonlinear Klein–Gordon and Sine-Gordon Equations


This chapter deals with the theory and applications of nonlinear Klein–Gordon (KG) and sine-Gordon (SG) equations. Special emphasis is given to various methods of solutions of these equations. The Green function method combined with integral transforms is employed to solve the linear Klein–Gordon equation. The Whitham averaging procedure and the Whitham averaged Lagrangian principle are used to discuss solutions of the nonlinear Klein–Gordon equation. Included are different ways of finding general and particular solutions of the sine-Gordon equation. Special attention is given to solitons, antisolitons, breather solutions and the energy associated with them, interaction of solitons, Bäcklund transformations, similarity solutions, and the inverse scattering method. Significant features of these methods and solutions are described with other ramifications.


Solitary Wave Wave Solution Soliton Solution Gordon Equation Solitary Wave Solution 
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  1. Biondini, G. and Wang, D. (2010). On the soliton solutions of the two-dimensional Toda Lattice, J. Phys. A, Math. Theor. 43(43), 1–20. #43007 MathSciNetCrossRefGoogle Scholar
  2. Collins, M.A. (1981). A quasi-continuum approach for solitons in an atomic chain, Chem. Phys. Lett. 77, 342–347. MathSciNetCrossRefGoogle Scholar
  3. Davis, H.T. (1962). Introduction to Nonlinear Differential and Integral Equations, Dover, New York. MATHGoogle Scholar
  4. De Leonardis, R.M. and Trullinger, S.E. (1980). Theory of boundary effects on sine-Gordon solitons, J. Appl. Phys. 51, 1211–1226. CrossRefGoogle Scholar
  5. Debnath, L. (1995). Integral Transforms and Their Applications, CRC Press, Boca Raton. MATHGoogle Scholar
  6. Debnath, L. and Bhatta, D. (2004). On solutions to a few linear fractional inhomogeneous partial differential equations in fluid mechanics, Fract. Calc. Appl. Anal. 7, 21–36. MathSciNetMATHGoogle Scholar
  7. Drazin, P.G. (1983). Solitons, Cambridge University Press, Cambridge. MATHCrossRefGoogle Scholar
  8. Duncan, D.B., Eilbeck, J.C., Feddersen, H., and Wattis, J.A.D. (1993). Solitons on Lattices, Physica D 68, 1–11. MATHCrossRefGoogle Scholar
  9. Dutta, M. and Debnath, L. (1965). Elements of the Theory of Elliptic and Associated Functions with Applications, World Press Pub. Ltd., Calcutta. MATHGoogle Scholar
  10. Eilbeck, J.C. and Flesch, B. (1990). Calculation of families of solitary waves on discrete lattices, Phys. Lett. A149, 200–202. MathSciNetGoogle Scholar
  11. Hirota, R. (1973a). Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys. 14, 805–809. MathSciNetMATHCrossRefGoogle Scholar
  12. Hirota, R. (1973b). Exact N-solutions of the wave equation of long waves in shallow water and in nonlinear lattices, J. Math. Phys. 14, 810–814. MathSciNetMATHCrossRefGoogle Scholar
  13. Lamb, G.L. (1971). Analytical descriptions of ultrashort optical pulse propagation in a resonant medium, Rev. Mod. Phys. 49, 99–124. MathSciNetCrossRefGoogle Scholar
  14. Lamb, G.L. (1973). Phase variation in coherent-optical-pulse propagation, Phys. Rev. Lett. 31, 196–199. CrossRefGoogle Scholar
  15. Lamb, G.L. (1980). Elements of Soliton Theory, Wiley, New York. MATHGoogle Scholar
  16. McCall, S.L. and Hahn, E.L. (1967). Self-induced transparency by pulsed coherent light, Phys. Rev. Lett. 18, 908–911. CrossRefGoogle Scholar
  17. McCall, S.L. and Hahn, E.L. (1969) Self-induced transparency, Phys. Rev. 183, 457–485. CrossRefGoogle Scholar
  18. Newell, A.C. and Kaup, D.J. (1978). Solitons as particles, oscillations, and in slowly changing media: A singular perturbation theory, Proc. R. Soc. Lond. A361, 413–446. Google Scholar
  19. Perring, J.K. and Skyrme, T.H.R. (1962). A model of unified field equation, Nucl. Phys. 31, 550–555. MathSciNetMATHCrossRefGoogle Scholar
  20. Scott, A.C. (1969). A nonlinear Klein–Gordon equation, Am. J. Phys. 37, 52–61. CrossRefGoogle Scholar
  21. Scott, A.C. (2003). Nonlinear Science, 2nd edition, Oxford University Press, Oxford. MATHGoogle Scholar
  22. Toda, M. (1967a). Vibration of a chain with nonlinear interaction, J. Phys. Soc. Jpn. 22, 431–436. CrossRefGoogle Scholar
  23. Toda, M. (1967b). Wave propagation in anharmonic lattices, J. Phys. Soc. Jpn. 23, 501–506. CrossRefGoogle Scholar
  24. Whitham, G.B. (1965a). A general approach to linear and nonlinear dispersive waves using a Lagrangian, J. Fluid Mech. 22, 273–283. MathSciNetCrossRefGoogle Scholar
  25. Whitham, G.B. (1965b). Nonlinear dispersive waves, Proc. R. Soc. Lond. A283, 238–261. MathSciNetGoogle Scholar
  26. Whitham, G.B. (1974). Linear and Nonlinear Waves, Wiley, New York. MATHGoogle Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas, Pan AmericanEdinburgUSA

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