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
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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.
W. Eckhaus, Studies in Nonlinear Stability Theory, Springer, Berlin, 1965
- G. Bard Ermentrout, Stable small-amplitude solutions in reaction-diffusion systems, Quart. Appl. Math. 39 (1981/82), no. 1, 61–86. MR 613952, DOI https://doi.org/10.1090/S0033-569X-1981-0613952-9
- L. N. Howard and N. Kopell, Slowly varying waves and shock structures in reaction-diffusion equations, Studies in Appl. Math. 56 (1976/77), no. 2, 95–145. MR 604035, DOI https://doi.org/10.1002/sapm197756295
- E. Knobloch and J. De Luca, Amplitude equations for travelling wave convection, Nonlinearity 3 (1990), no. 4, 975–980. MR 1079278
- Laurence R. Keefe, Dynamics of perturbed wavetrain solutions to the Ginzburg-Landau equation, Stud. Appl. Math. 73 (1985), no. 2, 91–153. MR 804366, DOI https://doi.org/10.1002/sapm198573291
- Y. Kuramoto, Chemical oscillations, waves, and turbulence, Springer Series in Synergetics, vol. 19, Springer-Verlag, Berlin, 1984. MR 762432
- Y. Kuramoto and S. Koga, Anomalous period-doubling bifurcations leading to chemical turbulence, Phys. Lett. A 92 (1982), no. 1, 1–4. MR 677805, DOI https://doi.org/10.1016/0375-9601%2882%2990725-3
- Charles G. Lange and Alan C. Newell, A stability criterion for envelope equations, SIAM J. Appl. Math. 27 (1974), 441–456. MR 381500, DOI https://doi.org/10.1137/0127034
- B. J. Matkowsky and Vl. A. Vol′pert, Coupled nonlocal complex Ginzburg-Landau equations in gasless combustion, Phys. D 54 (1992), no. 3, 203–219. MR 1146840, DOI https://doi.org/10.1016/0167-2789%2892%2990035-L
- H. T. Moon, P. Huerre, and L. G. Redekopp, Transitions to chaos in the Ginzburg-Landau equation, Phys. D 7 (1983), no. 1-3, 135–150. Order in chaos (Los Alamos, N.M., 1982). MR 719050, DOI https://doi.org/10.1016/0167-2789%2883%2990124-0
- J. E. Wesfreid, H. R. Brand, P. Manneville, G. Albinet, and N. Boccara (eds.), Propagation in systems far from equilibrium, Springer Series in Synergetics, vol. 41, Springer-Verlag, Berlin, 1988. MR 975039
- Alan C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38 (1969), no. 2, 279–303. MR 3363403, DOI https://doi.org/10.1017/S0022112069000176
- Paul K. Newton and Lawrence Sirovich, Instabilities of the Ginzburg-Landau equation: periodic solutions, Quart. Appl. Math. 44 (1986), no. 1, 49–58. MR 840442, DOI https://doi.org/10.1090/S0033-569X-1986-0840442-X
- Paul K. Newton and Lawrence Sirovich, Instabilities of the Ginzburg-Landau equation. II. Secondary bifurcation, Quart. Appl. Math. 44 (1986), no. 2, 367–374. MR 856192, DOI https://doi.org/10.1090/S0033-569X-1986-0856192-X
- Kazuhiro Nozaki and Naoaki Bekki, Chaos in a perturbed nonlinear Schrödinger equation, Phys. Rev. Lett. 50 (1983), no. 17, 1226–1229. MR 700325, DOI https://doi.org/10.1103/PhysRevLett.50.1226
- David O. Olagunju and Bernard J. Matkowsky, Burner-stabilized cellular flames, Quart. Appl. Math. 48 (1990), no. 4, 645–664. MR 1079911, DOI https://doi.org/10.1090/qam/1079911
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)
J. D. Rodriguez and L. Sirovich, Low dimensional dynamics for the complex Ginzburg-Landau equation, Physica 43D, 77–86 (1990)
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)
L. A. Segel, Distant side-walls cause slow amplitude modulation of cellular convection, J. Fluid Mech. 38, 203–224 (1969)
- Lawrence Sirovich and Paul K. Newton, Periodic solutions of the Ginzburg-Landau equation, Phys. D 21 (1986), no. 1, 115–125. MR 860011, DOI https://doi.org/10.1016/0167-2789%2886%2990082-5
- L. Sirovich, J. D. Rodriguez, and B. Knight, Two boundary value problems for the Ginzburg-Landau equation, Phys. D 43 (1990), no. 1, 63–76. MR 1060044, DOI https://doi.org/10.1016/0167-2789%2890%2990016-I
- K. Stewartson and J. T. Stuart, A non-linear instability theory for a wave system in plane Poiseuille flow, J. Fluid Mech. 48 (1971), 529–545. MR 309420, DOI https://doi.org/10.1017/S0022112071001733
J. T. Stuart and R. C. DiPrima, The Eckhaus and Benjamin-Feir resonance mechanisms, Proc. Roy. Soc. London Ser. A 362, 27–41 (1978)
W. Eckhaus, Studies in Nonlinear Stability Theory, Springer, Berlin, 1965
G. B. Ermentrout, Stable small-amplitude solutions in reaction-diffusion systems, Quart. Appl. Math. 39, 61–86 (1981)
L. N. Howard and N. Kopell, Slowly varying waves and shock structures in reaction-diffusion equations, Stud. Appl. Math. 56, 95–145 (1977)
E. Knobloch and J. DeLuca, Amplitude equations for travelling wave convection, Nonlinearity 3, 975–980 (1990)
L. R. Keefe, Dynamics of perturbed wavetrain solutions to the Ginzburg-Landau equation, Stud. Appl. Math. 73, 91–153 (1985)
Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, vol. 19, Springer Series in Synergetics, Springer-Verlag, Berlin and New York, 1984
Y. Kuramoto and S. Koga, Anomalous period-doubling bifurcations leading to chemical turbulence, Phys. Lett. 92A, 1–4 (1982)
C. G. Lange and A. C. Newell, A stability criterion for envelope equations, SIAM J. Appl. Math. 27, 441–456 (1974)
B. J. Matkowsky and V. Volpert, Coupled nonlocal complex Ginzburg-Landau equations in gasless combustion, Physica 54D, 203–219 (1992)
H. T. Moon, P. Huerre, and L. G. Redekopp, Transitions to chaos in the Ginzburg-Landau equation, Physica 7D, 135–150 (1983)
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
A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38, 279–303 (1969)
P. K. Newton and L. Sirovich, Instabilities of the Ginzburg-Landau equation: periodic solutions, Quart. Appl. Math. 44, 49–58 (1986)
P. K. Newton and L. Sirovich, Instabilities of the Ginzburg-Landau equation: Part II. Secondary bifurcation, Quart. Appl. Math 44, 367–374 (1986)
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)
D. O. Olagunju and B. J. Matkowsky, Burner stabilized cellular flames, Quart. Appl. Math. 48, 645–664 (1990)
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)
J. D. Rodriguez and L. Sirovich, Low dimensional dynamics for the complex Ginzburg-Landau equation, Physica 43D, 77–86 (1990)
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)
L. A. Segel, Distant side-walls cause slow amplitude modulation of cellular convection, J. Fluid Mech. 38, 203–224 (1969)
L. Sirovich and P. K. Newton, Periodic solutions of the Ginzburg-Landau equation, Physica 21D, 115–125 (1986)
L. Sirovich, J. D. Rodriguez, and B. Knight, Two boundary value problems for the Ginzburg-Landau equation, Physica 43D, 63–76 (1990)
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)
J. T. Stuart and R. C. DiPrima, The Eckhaus and Benjamin-Feir resonance mechanisms, Proc. Roy. Soc. London Ser. A 362, 27–41 (1978)
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© Copyright 1993
American Mathematical Society