Instabilities of the Ginzburg-Landau equation: periodic solutions

Newton, Paul K.; Sirovich, Lawrence

doi:10.1090/qam/840442

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

H. T. Moon, P. Huerre, and L. G. Redekopp, Three-frequency motion and chaos in the Ginzburg-Landau equation, Phys. Rev. Lett. 49 (1982), no. 7, 458–460. MR 665865, DOI https://doi.org/10.1103/PhysRevLett.49.458
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
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

Y. Kuramoto, Diffusion induced chaos in reaction systems, Supp. Prog. Theor. Phys., No. 64, 1978

C. S. Bretherton and E. A. Spiegel, Intermittency through modulational instability, Physics Letters, Vol. 96A, #3, 20 June 1983

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
Robert J. Deissler, Noise-sustained structure, intermittency, and the Ginzburg-Landau equation, J. Statist. Phys. 40 (1985), no. 3-4, 371–395. MR 806709, DOI https://doi.org/10.1007/BF01017180
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
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

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)

D. J. Benney and G. J. Roskes, Wave instabilities, Stud. Appl. Math. 48, 377 (1969)

A. Davey and K. Stewartson, On three-dimensional packets of surface waves, Proc. Roy. Soc. London Ser. A 338 (1974), 101–110. MR 349126, DOI https://doi.org/10.1098/rspa.1974.0076

T. B. Benjamin and J. E. Feir, The disintegration of wave trains on deep water, J. Fluid Mech. 27, 417 (1967)

H. Hasimoto and H. Ono, Nonlinear modulation of gravity waves, J. Phys. Soc. Japan 33, 805 (1972)

Alan Jeffrey and Takuji Kawahara, Asymptotic methods in nonlinear wave theory, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. Applicable Mathematics Series. MR 646264
G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0483954
Wiktor Eckhaus, Studies in non-linear stability theory, Springer Tracts in Natural Philosophy, Vol. 6, Springer-Verlag New York, New York, Inc., 1965. MR 0196298

J. T. Stuart and R. C. DiPrima, The Eckhaus and Benjamin-Feir resonance mechanisms, Proc. Roy. Soc. Lond. A 362, 27 (1978)

A. Thyagaraja, Recurrence phenomena and the number of effective degrees of freedom in nonlinear wave motions, Nonlinear waves (Greenville, N.C., 1982) Cambridge Monogr. Mech. Appl. Math., Cambridge Univ. Press, Cambridge, 1983, pp. 308–325. MR 744063, DOI https://doi.org/10.1017/CBO9780511569500.018
A. Thyagaraja, Recurrence, dimensionality, and Lagrange stability of solutions of the nonlinear Schrödinger equation, Phys. Fluids 24 (1981), no. 11, 1973–1975. MR 636057, DOI https://doi.org/10.1063/1.863294
V. V. Nemytskii and V. V. Stepanov, Qualitative theory of differential equations, Princeton Mathematical Series, No. 22, Princeton University Press, Princeton, N.J., 1960. MR 0121520