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Global spatially periodic solutions to the Ginzburg–Landau equation

Published online by Cambridge University Press:  14 November 2011

Yisong Yang
Yisong Yang
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, U.S.A.
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Synopsis

In this paper we study the global existence and nonexistence of spatially periodic solutions to the initial value problem of the Ginzburg–Landau equation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 110 , Issue 3-4 , 1988 , pp. 263 - 273
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

1Baillon, J. B., Cazenave, T. and Figueira, M.. Équation de Schrödinger non linéaire. C.R. Acad. Sci. Paris Sér. A 284 (1977), 869–872.Google Scholar
2Keefe, L. R.. Dynamics of perturbed wavetrain solutions to the Ginzburg–Landau equation. Stud. Appl. Math. 73 (1985), 91–153.CrossRefGoogle Scholar
3Kogelman, S. and DiPrima, R. C.. Stability of spatially periodic supercritical flows in hydrodynamics. Phys. Fluids 13 (1970), 1–11.Google Scholar
4Kuramoto, Y. and Yamada, T.. Turbulent state in chemical reactions. Progr. Theoret. Phys. 56 (1976), 679–681.Google Scholar
5Moon, H. T., Huerre, P. and Redekopp, L. G.. Three-frequency motion and chaos in the Ginzburg–Landau equation. Phys. Rev. Lett. 49 (1982), 458–460.CrossRefGoogle Scholar
6Moon, H. T., Huerre, P. and Redekopp, L. G.. Transitions to chaos in the Ginzburg–Landau equation. Physica D 7 (1983), 135–150.CrossRefGoogle Scholar
7Newell, A. C. and Whitehead, J. A.. Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38 (1969), 279–303.Google Scholar
8Newton, P. K. and Sirovich, L.. Instabilities of the Ginzburg–Landau equation: periodic solutions. Quart. Appl. Math. 44 (1986), 49–58.Google Scholar
9Nozaki, K. and Bekki, N.. Pattern selection and spatiotemporal transition to chaos in the Ginzburg–Landau equation. Phys. Rev. Lett. 51 (1983), 2171–2174.Google Scholar
10Sattinger, D. H.. Topics in Stability and Bifurcation Theory. Lecture Notes in Mathematics 309 (New York: Springer, 1972).Google Scholar
11Showalter, R. E.. Hilbert Space Methods for Partial Differential Equations (San Francisco: Pitman, 1979).Google Scholar
12Sirovich, L. and Newton, P. K.. Periodic solutions of the Ginzburg–Landau equation. Physica D 21 (1986), 115–125.Google Scholar
13Thyagaraja, A.. Recurrent motions in certain continuum dynamical systems. Phys. Fluids 22 (1979), 2093–2096.CrossRefGoogle Scholar
14Thyagaraja, A.. Recurrence, dimensionality, and Lagrange stability of solutions of the nonlinear Schrödinger equation. Phys. Fluids 24 (1981), 1973–1975.Google Scholar
15Yang, Y.. On the Lagrange stability of spatially periodic solutions of the Ginzburg–Landau equation. Quart. Appl. Math, (to appear).Google Scholar