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Asymptotic Evaluation Methods of Nonlinear Differential Equations Near the Instability Point

  • Masuo Suzuki
Part of the NATO Advanced Study Institutes Series book series (NSSB, volume 75)

Abstract

The purpose of this paper is to present a general method [Suzuki, 1976a, b, 1977a, b, c, 1978, 1980, in press, to be published] to evaluate asymptotically solutions of nonlinear stochastic differential equations or Langevin’s equations when the system is initially located at or near the unstable point. One of our keypoints is to notice the existence of the scaling regime in an intermediate time region, in which the temporal evolution of physical quantities is expressed by a certain scaling function of the so-called scaling variable τ = S(t, ɛ, …) of time t and the relevant smallness parameter ɛ. Namely, we divide the whole time region into three regimes, i.e., initial, second (scaling), and final regimes.

Keywords

Nonlinear Differential Equation Nonlinear Transformation Transient Phenomenon Instability Point Final Regime 
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References

  1. Caroli, B., Caroli, C. and Roulet, B., 1979, Diffusion in a bistable potential: a systematic WKB treatment, J. Stat. Phys. 21: 415.ADSCrossRefGoogle Scholar
  2. Caroli, B., Caroli, C. and Roulet, B., 1980, Growth of fluctuations from a marginal equilibrium, Physica, 101A: 581.MathSciNetCrossRefGoogle Scholar
  3. de Pasquale, F., Tartaglia, P., and Tombesi, P., 1979, Transient laser radiation as a stochastic process near an instability point, Physica, 99A: 581.CrossRefGoogle Scholar
  4. Suzuki, M., 1976a, Scaling theory of nonequilibrium systems near the instability point. I. General aspects of transient phenomena, Prog. Theor. Phys., 56: 77.ADSMATHCrossRefGoogle Scholar
  5. Suzuki, M., 1976b, Scaling theory of nonequilibrium systems near the instability point. II. Anomalous fluctuation theorems in the extensive region, Prog. Theor. Phys., 56: 477.ADSCrossRefGoogle Scholar
  6. Suzuki, M., 1977a, Scaling theory of nonequilibrium systems near the instability point. III. Continuation to find region and systematic scaling expansion, Prog. Theor. Phys., 57: 380.ADSCrossRefGoogle Scholar
  7. Suzuki, M., 1977b, Scaling theory of transient phenomena near the instability point, J. Stat. Phys., 16: 11.ADSCrossRefGoogle Scholar
  8. Suzuki, M., 1980, Advances in Chem. Phys., 46: 195.Google Scholar
  9. Suzuki, M., in press, in: “Proceedings of the XVII Solvay Conference on Physics”, John Wiley and Sons, Inc., N.Y. Suzuki, M., to be published, in: “Lecture Notes in Physics. Proceedings of the Vlth Sites Conference on Statistical Mechanics”, Springer-Verlag, Berlin.Google Scholar

Copyright information

© Plenum Press, New York 1981

Authors and Affiliations

  • Masuo Suzuki
    • 1
  1. 1.Department of PhysicsUniversity of TokyoHongo, Bunkyo-ku TokyoJapan

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