Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Instabilities of the Ginzburg-Landau equation: periodic solutions

Authors: Paul K. Newton and Lawrence Sirovich
Journal: Quart. Appl. Math. 44 (1986), 49-58
MSC: Primary 35Q20; Secondary 58F21
DOI: https://doi.org/10.1090/qam/840442
MathSciNet review: 840442
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Abstract: The evolution of spatially periodic unstable solutions to the Ginzburg-Landau equation is considered. These solutions are shown to remain pointwise bounded (Lagrange stable). The first step in the route to chaos is limit cycle behavior. This is treated by perturbation theory and shown to result in a factorable form. Agreement between the perturbation result and an exact numerical integration is shown to be excellent.

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  • [1] H. T. Moon, P. Huerre, L. G. Redekopp, Three-frequency motion and chaos in the Ginzburg-Landau equation, Phys. Rev. Letters, Vol. 49, #7, 16 Aug. 1982 MR 665865
  • [2] H. T. Moon, P. Huerre, L. G. Redekopp, Transitions to chaos in the Ginzburg-Landau equation, Physica 7D, 135 (1983) MR 719050
  • [3] K. Nozaki and N. Bekki, Pattern selection and spatiotemporal transition to chaos in the Ginzburg-Landau equation, Phys. Rev. Letters, Vol. 51, #24, 12 Dec. 1983 MR 700325
  • [4] Y. Kuramoto, Diffusion induced chaos in reaction systems, Supp. Prog. Theor. Phys., No. 64, 1978
  • [5] C. S. Bretherton and E. A. Spiegel, Intermittency through modulational instability, Physics Letters, Vol. 96A, #3, 20 June 1983
  • [6] L. Keefe, Dynamics of perturbed wavetrain solutions to the Ginzburg-Landau equation, Ph.D. Dissertation, University of Southern California, 1984 MR 804366
  • [7] R. J. Deissler, Noise-sustained structure intermittency and the Ginzburg-Landau equation, submitted for publication. MR 806709
  • [8] A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38, 279 (1969) MR 3363403
  • [9] K. Stewartson and J. T. Stuart, A non-linear instability theory for a wave system in plane Poiseuille flow, J. Fluid Mech. 48, 529 (1971) MR 0309420
  • [10] R. C. DiPrima, W. Eckhaus, and L. A. Segel, Non-linear wave-number interaction in near-critical two dimensional flows, J. Fluid Mech. 49, 705 (1971)
  • [11] D. J. Benney and G. J. Roskes, Wave instabilities, Stud. Appl. Math. 48, 377 (1969)
  • [12] A. Davey and K. Stewartson, On three-dimensional packets of surface waves, Proc. Roy. Soc. London A338, 101 (1974) MR 0349126
  • [13] T. B. Benjamin and J. E. Feir, The disintegration of wave trains on deep water, J. Fluid Mech. 27, 417 (1967)
  • [14] H. Hasimoto and H. Ono, Nonlinear modulation of gravity waves, J. Phys. Soc. Japan 33, 805 (1972)
  • [15] A. Jeffrey and T. Kawahara, Asymptotic methods in nonlinear wave theory, Pittman, 1982 MR 646264
  • [16] G. B. Whitham, Linear and nonlinear waves, John Wiley & Sons, Inc., 1974 MR 0483954
  • [17] W. Eckhaus, Studies in non-linear stability theory, Springer Tracts in Natural Philosophy, vol. 6, Springer-Verlag, 1965 MR 0196298
  • [18] J. T. Stuart and R. C. DiPrima, The Eckhaus and Benjamin-Feir resonance mechanisms, Proc. Roy. Soc. Lond. A 362, 27 (1978)
  • [19] A. Thyagaraja, in Nonlinear waves, ed. L. Debnath, Chapter 17: ``Recurrence Phenomena and the Number of Effective Degrees of Freedom in Nonlinear Wave Motions,'' Cambridge, 1983 MR 744063
  • [20] A. Thyagaraja, Recurrence, dimensionality, and Lagrange stability of solutions of the nonlinear Schrödinger equation, Phys. Fluids 24 (11), Nov. 1981 MR 636057
  • [21] V. V. Nemytiskii and V. V. Stepanov, Qualitative theory of differential equations, Princeton Univ. Press, 1960 MR 0121520

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DOI: https://doi.org/10.1090/qam/840442
Article copyright: © Copyright 1986 American Mathematical Society

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