J. Phys. Soc. Jpn. 34, pp. 1073-1082 (1973) [10 Pages]

On the Nonlinear Schrödinger Equation for Langmuir Waves

+ Affiliations
1Physics Department, University of California2Department of Physics and Atomic Energy Research Institute, College of Science and Engineering, Nihon University

A direct derivation of the nonlinear schrödinger equation for Langmuir waves is presented, based upon the nonlinear wave packet ansatz of Karpman and Krushkal. Both fluid and Vlasov equation formulations are used. The results obtained are essentially equivalent to those found earlier by Taniuti, et al. using reductive perturbation theory, including the importance of wave particle resonances at the group velocity for the long time behavior of the amplitude of modulated waves. Separating the wave packet considerations from the calculation of the nonlinear frequency shift makes it possible to attack the latter with whatever method facilitates the analysis of that part of the problem. In addition, certain ambiguities concerning singularities in velocity integrations are resolved, and the connection with a well-posed initial value problem is made somewhat clearer. This method can be used equally well for other waves, and may be of help particularly in situations where it is not clear, a priori, what scaling to adopt in applying reductive perturbation theory.

©1973 The Physical Society of Japan


  • 1 C. S.Gardner and G. K.Morikawa: Rep. NYU-9082, Courant Inst, of Math. Sci., New York Univ., New York, 1960; Google ScholarH.Washimi and T.Taniuti: Phys. Rev. Letters 17 (1966) 966; T.Taniuti and C. C.Wei: J. Phys. Soc. Japan 24 (1968) 941; Crossref;, Google ScholarA brief review and farther references are given in & #x201C;Methods in Non-Linear Plasma Theory & #x201D; Google ScholarR. C.Davidson, Charp. 2 (Academic Press, New York, 1972). Google Scholar
  • 2 T.Taniuti and N.Yajima: J. math. Phys. 10 (1969) 1369; K.Shimizu and Y. H.Ichikawa: J. Phys. Soc. Japan 33 (1972) 789. CrossrefGoogle Scholar
  • 3 G. B.Whitham: Proc. Roy. Soc. A 283 (1965) 238: J. Fluid Mech. 22 (1965) 273; M. J.Lighthill: Proc. Roy. Soc. A 299 (1967) 28. CrossrefGoogle Scholar
  • 4 N.Asano, T.Taniuti and N.Yajima: J. math. Phys. 10 (1969) 2020; H.Washimi and T.Taniuti: Phys. Rev. Letters 17 (1966) 996. CrossrefGoogle Scholar
  • 5 V. I.Karpman and E. M.Krushkal: Zh. Eksp. Teor. Fiz. 55 (1968) 530. translation: Soviet Physics & #x2013;JETP 28 (1969) 277. Google Scholar
  • 6 Y. H.Ichikawa, T.Imamura and T.Taniuti: J. Phys. Soc. Japan 33 (1972) 189. LinkGoogle Scholar
  • 7 Y. H.Ichikawa and T.Taniuti: J. Phys. Soc. Japan 34 (1973) 513. LinkGoogle Scholar
  • 8 B. B.Kadomtsev: Plasma Turbulence (Academic Press, London, 1965). Google Scholar
  • 9 A. L.Brinca: & #x201CModulational Instability of Whistlers in Cold Plasmas & #x201D; Stanford Univ. IPR Report 464 (1972). Google Scholar
  • 10 N.Kryloff and N.Bogoliuboff: Introduction to Nonlinear Mechanics (Princeton Univ. Press, 1947). Google Scholar
  • 11 This calculation has been carried out by Y.Midzuno (private communication). Google Scholar
  • 12 We sketch the algebraic details here, since in contrast to the conventional treatments, designed for turbulence problems, we do not take phase or ensemble averages. This leads to some small formal differences in the analysis. Google Scholar
  • 13 Yoshi H.Ichikawa, T.Suzuki and T.Taniuti: & #x201C;Modulation Instability of Electron Plasma Wave & #x201D; Nihon University, Preprint NUP-A-72-17. Google Scholar
  • 14 I. B.Bernstein, J. M.Greene and M. D.Krushal: Phys. Rev. 108 (1947) 546. CrossrefGoogle Scholar