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A note on the Lagrangian method for nonlinear dispersive waves

Published online by Cambridge University Press:  13 March 2009

T. Kawahara
T. Kawahara
Affiliation:
Department of Physics, Faculty of Science, Kyoto University, Kyoto 606, Japan
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Abstract

This paper makes a few remarks on the method of the averaged Lagrangian developed by Whitham to describe slow variations of nonlinear wave trains. The concept of multiple scales is incorporated into the variational formalism, and a consistent development to higher approximation is suggested as a formal perturbation based on the variational principle. The propagation of weakly dispersive long waves is also reconsidered in relation to the Lagrangian method. It is demonstrated that the nonlinear Schrodinger equation and the Korteweg–de Vries equation can be derived from the Euler–Lagrange equations of the perturbed Lagrangian.

Type
Research Article
Information
Journal of Plasma Physics , Volume 18 , Issue 2 , October 1977 , pp. 305 - 316
Copyright
Copyright © Cambridge University Press 1977

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References

REFERENCES

Benney, D. J. & Newell, A. C. 1967 J. Math. Phys. 46, 133.CrossRefGoogle Scholar
Bretherton, F. P. & Garrett, C. J. R. 1969 Proc. Roy. Soc. A 302, 529.Google Scholar
Davey, A. 1972 J. Fluid Mech. 53, 796.Google Scholar
Dewar, R. L. 1970 Phys. Fluids, 13, 2710.CrossRefGoogle Scholar
Dewar, R. L. 1972 J. Plasma Phys. 7, 267.CrossRefGoogle Scholar
Dougherty, J. P. 1970 J. Plasma Phys. 4, 761.CrossRefGoogle Scholar
Dougherty, J. P. 1974 J. Plasma Phys. 11, 331.CrossRefGoogle Scholar
Dysthe, K. B. 1974 J. Plasma Phys. 11, 63.CrossRefGoogle Scholar
Galloway, J. J. & Kim, H. 1971 J. Plasma Phys. 6, 53.CrossRefGoogle Scholar
Hayes, W. D. 1973 Proc. Roy. Soc. A 332, 199.Google Scholar
Kakutani, T. & Sugimoto, N. 1974 Phys. Fluids, 17, 1617.CrossRefGoogle Scholar
Karpman, V. I. & Krushkal, E. M. 1969 Soviet, Phys. JETP, 28, 277.Google Scholar
Kawahra, T. 1973 J. Phys. Soc. Japan, 35, 1537.CrossRefGoogle Scholar
Kawahara, T. 1975 J. Phys. Soc. Japan, 38, 1200.CrossRefGoogle Scholar
Luke, J. C. 1966 Proc. Roy. Soc. A 292, 403.Google Scholar
Nayfeh, A. H. 1973 Perturbation Methods. Wiley.Google Scholar
Sandri, G. 1963 Ann. Phys. 24, 332, 380.CrossRefGoogle Scholar
Sandri, G. 1965 Nuovo Cimento, 36, 67.CrossRefGoogle Scholar
Sturrock, P. A. 1957 Proc. Roy. Soc. A 242, 277.Google Scholar
Taniuti, T. & Yajima, N. 1969 J. Math. Phys. 10, 1369.CrossRefGoogle Scholar
Taniuti, T. 1974 Prog. Theor. Phys. Suppl. 55.Google Scholar
Whitham, G. B. 1965 a Proc. Roy. Soc. A 283, 238.Google Scholar
Whitham, G. B. 1965 b J. Fluid Mech. 22, 273.CrossRefGoogle Scholar
Whitham, G. B. 1967 a Proc. Roy. Soc. A 299, 6.Google Scholar
Whitham, G. B. 1967 b J. Fluid Mech. 27, 399.CrossRefGoogle Scholar
Whitham, G. B. 1970 J. Fluid Mech. 44, 373.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
Willebrand, J. 1975 J. Fluid Mech. 70, 113.CrossRefGoogle Scholar