Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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

   
 
 

 

Spatial heteroclinic bifurcation of time periodic solutions to the Ginzburg-Landau equation


Authors: H. Dang-Vu and C. Delcarte
Journal: Quart. Appl. Math. 58 (2000), 459-472
MSC: Primary 35Q55; Secondary 34C37, 35B10, 37D45, 37L10
DOI: https://doi.org/10.1090/qam/1770649
MathSciNet review: MR1770649
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the spatial structure of the time periodic solutions to the Ginzburg-Landau equation in various configurations (supercritical and subcritical). We show that spatially periodic bursting solutions or spatially heteroclinic solutions can occur depending upon the values of the coefficients. As a consequence of this study, we obtain an exact solution to the nonlinear system of Kapitula and Maier-Paape [16]. We then show that near a spatially heteroclinic solution there is an extremely intricate complex of spatially unstable solutions and KAM surfaces.


References [Enhancements On Off] (What's this?)

  • [1] V. I. Arnold, Mathematical Methods in Classical Mechanics, Springer-Verlag, New York, 1978 MR 0690288
  • [2] G. D. Birkhoff, Nouvelles recherches sur les systèmes dynamiques, Mem. Pont. Acad. Sci. Novi Lyncei 1, 85-216 (1935)
  • [3] J. Bricmont and A. Kupiainen, Stability of moving fronts in the Ginzburg-Landau equation, Comm. Math. Phys. 159, 287-318 (1994) MR 1256990
  • [4] F. Cariello and M. Tabor, Painlevé expansions for nonintegrable evolution equations, Physica D 39, 77-94 (1989) MR 1021183
  • [5] H. Dang-Vu and C. Delcarte, Bifurcations and chaos in the steady-state solutions of the Ginzburg-Landau equation in: ``International Conference on Nonlinear Dynamics, Chaotic and Complex Systems", Zakopane, Poland, 7-12 November, 1995, Polish Scientific Publishers PWN, Warszawa, 1996, pp. 317-325 MR 1443466
  • [6] A. Doelman, Slow time-periodic solutions of the Ginzburg-Landau equation, Physica D 40, 156-172 (1989) MR 1029461
  • [7] A. Doelman, Finite-dimensional models of the Ginzburg-Landau equation, Nonlinearity 4, 231-250 (1991) MR 1107006
  • [8] A. Doelman, Traveling waves in the complex Ginzburg-Landau equation, J. Nonlinear Sci., 3, 225-266 (1993) MR 1220175
  • [9] A. Doelman, Breaking the hidden symmetry in the Ginzburg-Landau equation, Physica D 97, 398-428 (1996) MR 1412551
  • [10] C. R. Doering, J. D. Gibbon, D. D. Holm, and B. Nicolaenko, Low dimensional behaviour in the complex Ginzburg-Landau equations, Nonlinearity 1, 279-309 (1988) MR 937004
  • [11] W. Eckhaus, The Ginzburg-Landau manifold is an attractor, J. Nonlinear Sci. 3, 329-348 (1993) MR 1237095
  • [12] J.-P. Eckmann and Th. Gallay, Front solutions for the Ginzburg-Landau equation, Comm. Math. Phys. 152, 221-248 (1993) MR 1210167
  • [13] M. Hénon, The applicability of the third integral of motion: Some numerical experiments, Astrophys. J. 69, 73-79 (1964) MR 0158746
  • [14] L. M. Hocking and K. Stewartson, On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance, Proc. Roy. Soc. London 326A, 289-313 (1972) MR 0329419
  • [15] Ph. Holmes, Spatial structure of time-periodic solutions of the Ginzburg-Landau equation, Physica D 23, 84-90 (1986) MR 876910
  • [16] T. Kapitula and S. Maier-Paape, Spatial dynamics of time periodic solutions for the Ginzburg-Landau equation, Z. angew. Math. Phys. (ZAMP) 47, 265-305 (1996) MR 1385917
  • [17] L. R. Keefe, Dynamics of perturbed wavetrain solutions to Ginzburg-Landau equation, Stud. Appl. Math. 73, 91-153 (1985) MR 804366
  • [18] M. Landman, Solutions of the Ginzburg-Landau equation of interest in shear flow transition, Stud. Appl. Math. 76, 187-237 (1987) MR 1039301
  • [19] B. J. Matkowsky and V. Volpert, Stability of plane wave solutions of complex Ginzburg-Landau equations, Quart. Appl. Math. 51, 265-281 (1993) MR 1218368
  • [20] H. T. Moon, P. Huerre, and L. G. Redekopp, Transitions to chaos in the Ginzburg-Landau equation, Physica D 7, 135-152 (1983) MR 719050
  • [21] P. K. Newton and L. Sirovich, Instabilities of the Ginzburg-Landau equation: Periodic solutions, Quart. Appl. Math. 44, 49-58 (1986) MR 840442
  • [22] K. Nozaki and N. Bekki, Exact solutions of the generalized Ginzburg-Landau equation, J. Phys. Soc. Jap. 53, 1581-1582 (1984) MR 750609
  • [23] B. I. Shraiman, A. Pumir, W. van Saarloos, P. C. Hohenberg, H. Chaté, and M. Holen, Spatiotemporal chaos in the one-dimensional complex Ginzburg-Landau equation, Physica D 57, 241-248 (1992) MR 1182049
  • [24] L. Sirovich and P. K. Newton, Periodic solutions of the Ginzburg-Landau equation, Physica D 21, 115-125 (1986) MR 860011
  • [25] P. Takáč, Invariant 2-tori in the time-dependent Ginzburg-Landau equation, Nonlinearity 5, 289-321 (1992) MR 1158376
  • [26] W. van Saarloos and P. C. Hohenberg, Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations, Physica D 56, 303-367 (1992) MR 1169610
  • [27] Y. Yang, Global spatially periodic solutions to the Ginzburg-Landau equation, Proc. Roy. Soc. Edinburgh A 110, 263-273 (1988) MR 974742

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 35Q55, 34C37, 35B10, 37D45, 37L10

Retrieve articles in all journals with MSC: 35Q55, 34C37, 35B10, 37D45, 37L10


Additional Information

DOI: https://doi.org/10.1090/qam/1770649
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society