Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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

   
 
 

 

Stability of plane wave solutions of complex Ginzburg-Landau equations


Authors: B. J. Matkowsky and Vl. A. Volpert
Journal: Quart. Appl. Math. 51 (1993), 265-281
MSC: Primary 35Q55; Secondary 35B35, 76E30
DOI: https://doi.org/10.1090/qam/1218368
MathSciNet review: MR1218368
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the stability of plane wave solutions of both single and coupled complex Ginzburg-Landau equations and determine stability domains in the space of coefficients of the equations.


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

  • [1] W. Eckhaus, Studies in Nonlinear Stability Theory, Springer, Berlin, 1965
  • [2] G. B. Ermentrout, Stable small-amplitude solutions in reaction-diffusion systems, Quart. Appl. Math. 39, 61-86 (1981) MR 613952
  • [3] L. N. Howard and N. Kopell, Slowly varying waves and shock structures in reaction-diffusion equations, Stud. Appl. Math. 56, 95-145 (1977) MR 0604035
  • [4] E. Knobloch and J. DeLuca, Amplitude equations for travelling wave convection, Nonlinearity 3, 975-980 (1990) MR 1079278
  • [5] L. R. Keefe, Dynamics of perturbed wavetrain solutions to the Ginzburg-Landau equation, Stud. Appl. Math. 73, 91-153 (1985) MR 804366
  • [6] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, vol. 19, Springer Series in Synergetics, Springer-Verlag, Berlin and New York, 1984 MR 762432
  • [7] Y. Kuramoto and S. Koga, Anomalous period-doubling bifurcations leading to chemical turbulence, Phys. Lett. 92A, 1-4 (1982) MR 677805
  • [8] C. G. Lange and A. C. Newell, A stability criterion for envelope equations, SIAM J. Appl. Math. 27, 441-456 (1974) MR 0381500
  • [9] B. J. Matkowsky and V. Volpert, Coupled nonlocal complex Ginzburg-Landau equations in gasless combustion, Physica 54D, 203-219 (1992) MR 1146840
  • [10] H. T. Moon, P. Huerre, and L. G. Redekopp, Transitions to chaos in the Ginzburg-Landau equation, Physica 7D, 135-150 (1983) MR 719050
  • [11] A. C. Newell, Dynamics of patterns: A survey, Propagation in Systems Far from Equilibrium, Proceedings of Les Houches Workshop (J. E. Wesfried, H. R. Brand, P. Manneville, G. Albinet, and N. Boccara, eds.), Springer-Verlag, Berlin, Heidelberg, 122-155, 1987 MR 975039
  • [12] A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38, 279-303 (1969) MR 3363403
  • [13] P. K. Newton and L. Sirovich, Instabilities of the Ginzburg-Landau equation: periodic solutions, Quart. Appl. Math. 44, 49-58 (1986) MR 840442
  • [14] P. K. Newton and L. Sirovich, Instabilities of the Ginzburg-Landau equation: Part II. Secondary bifurcation, Quart. Appl. Math 44, 367-374 (1986) MR 856192
  • [15] K. Nozaki and N. Bekki, Pattern selection and spatiotemporal transition to chaos in the Ginzburg-Landau equation, Phys. Rev. Lett. 51, No. 24, 2171-2174 (1983) MR 700325
  • [16] D. O. Olagunju and B. J. Matkowsky, Burner stabilized cellular flames, Quart. Appl. Math. 48, 645-664 (1990) MR 1079911
  • [17] D. O. Olagunju and B. J. Matkowsky, Coupled complex Ginzburg-Landau type equations in gaseous combustion, Stability and Applied Analysis of Continuous Media 2, No. 1, 1-27 (1992)
  • [18] J. D. Rodriguez and L. Sirovich, Low dimensional dynamics for the complex Ginzburg-Landau equation, Physica 43D, 77-86 (1990)
  • [19] W. Schöpf and L. Kramer, Small-amplitude periodic and chaotic solutions of the complex Ginzburg-Landau equation for a subcritical bifurcation, Phys. Rev. Lett. 66, 2316-2319 (1991)
  • [20] L. A. Segel, Distant side-walls cause slow amplitude modulation of cellular convection, J. Fluid Mech. 38, 203-224 (1969)
  • [21] L. Sirovich and P. K. Newton, Periodic solutions of the Ginzburg-Landau equation, Physica 21D, 115-125 (1986) MR 860011
  • [22] L. Sirovich, J. D. Rodriguez, and B. Knight, Two boundary value problems for the Ginzburg-Landau equation, Physica 43D, 63-76 (1990) MR 1060044
  • [23] K. Stewartson and J. T. Stuart, A non-linear instability theory for a wave system in plane Poiseuille flow, J. Fluid Mech. 48, 529-545 (1971) MR 0309420
  • [24] J. T. Stuart and R. C. DiPrima, The Eckhaus and Benjamin-Feir resonance mechanisms, Proc. Roy. Soc. London Ser. A 362, 27-41 (1978)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 35Q55, 35B35, 76E30

Retrieve articles in all journals with MSC: 35Q55, 35B35, 76E30


Additional Information

DOI: https://doi.org/10.1090/qam/1218368
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society