The three-wave Interaction of four waves Revisited: A Lax Pair and Possibly General Solution

  • Filipe J. Romeiras
Part of the NATO ASI Series book series (NSSB, volume 331)


The nonlinear resonant interaction of coherent waves is a fundamental process in the study of wave phenomena which has received a great deal of attention in its many aspects (Kaup et al., 1979; Craik, 1985; Ablowitz and Clarkson, 1991; and references therein).


Hamiltonian System Elliptic Function Laurent Series Explosive Instability Autonomous Ordinary Differential Equation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ablowitz, M.A., and Clarkson, P.A., 1991, “Solitons, Nonlinear Evolution Equations and Inverse Scattering”, Cambridge University Press, Cambridge.CrossRefzbMATHGoogle Scholar
  2. Aristov, V.V., Karplyuk, K.S., and Pavlenko, V.P., 1972, Coupling effects of three-wave interaction on development of explosive instability, Ukrain. Phys. J. 17: 307.Google Scholar
  3. Arnold, V.I., 1978, “Mathematical Methods of Classical Mechanics”, Springer-Verlag, New York.CrossRefzbMATHGoogle Scholar
  4. Craik, A.D.D., 1985, “Wave Interactions and Fluid Flows”, Cambridge University Press, Cambridge.zbMATHGoogle Scholar
  5. Davidson, R.C., 1972, “Methods in Nonlinear Plasma Theory”, Academic Press, New York.Google Scholar
  6. Goldstein, H., 1980, “Classical Mechanics”, Addison-Wesley, Reading, Massachusetts.zbMATHGoogle Scholar
  7. Karplyuk, K.S., Oraevskii, V.N., and Pavlenko, V.P., Dynamics of the nonlinear interaction of MHD waves, Plasma Phys. 15: 113.Google Scholar
  8. Kaup, D.J., Reiman, A., and Bers, A., 1979, Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium, Rev. Mod. Phys. 51: 275.CrossRefMathSciNetADSGoogle Scholar
  9. Lax, P.D., 1968, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math. 21: 467.CrossRefzbMATHMathSciNetGoogle Scholar
  10. Ramani, A., Grammaticos, B., and Bountis, T., The Painlevé property and singularity analysis of integrable and non-integrable systems, Phys. Rep. 180: 159.Google Scholar
  11. Rauch-Wojciechowski, S., 1990, Hamiltonian structures and complete integrability in analytical mechanics, in “Soliton Theory: A Survey of Results”, A.P. Fordy, ed., Manchester University Press, Manchester.Google Scholar
  12. Romeiras, F.J., 1983, Integrability of double three-wave interaction, Phys. Lett. A 93: 227.CrossRefMathSciNetADSGoogle Scholar
  13. Romeiras, F.J., 1993, to be published.Google Scholar
  14. Sugihara, R., 1968, Interaction between an electromagnetic wave, plasma waves and an ion acoustic waves, Phys. Fluids 11: 178.CrossRefADSGoogle Scholar
  15. Verheest, F., 1987, Non-linear wave interactions in a complex Hamiltonian formalism, J. Phys. A: Math. Gen. 20: 103.CrossRefzbMATHMathSciNetADSGoogle Scholar
  16. Verheest, F., 1988, Integrability of restricted multiple three-wave interactions. II., Coupling constants with ratios 1 and 2, J. Math. Phys. 29: 2197.CrossRefMathSciNetADSGoogle Scholar
  17. Walters, D., and Lewak, G.J., 1977, Dynamics of four coupled plasma waves to second order, J. Plasma Phys. 18: 525.CrossRefADSGoogle Scholar
  18. Wojciechowski, S., Jiang, Z., and Bullough, R.K., 1986, Integrable multiwave interaction systems of ODEs, Phys. Lett. A 117: 399.CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Filipe J. Romeiras
    • 1
  1. 1.Departamento de Matemática e Centro de ElectrodinâmicaInstituto Superior TécnicoLisboa CodexPortugal

Personalised recommendations