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On the Cauchy problem for the 1+2 complex Ginzburg-Landau equation

Published online by Cambridge University Press:  17 February 2009

Charles Bu
Charles Bu
Affiliation:
Department of Mathematics, Wellesley College, Wellesley, Massachusetts 02181, U.S.A.
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Abstract

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We present analytical methods to investigate the Cauchy problem for the complex Ginzburg-Landau equation u1 = (v + iα)Δu − (κ + iβ) |u|2qu + γu in 2 spatial dimensions (here all parameters are real). We first obtain the local existence for v > 0, κ ≥ 0. Global existence is established in the critical case q = 1. In addition, we prove the global existence when .

Type
Research Article
Information
The ANZIAM Journal , Volume 36 , Issue 3 , January 1995 , pp. 313 - 324
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Ablowitz, M. J. and Segur, H., Solitons and the inverse scattering transform (North-Holland, 1981).CrossRefGoogle Scholar
[2]Bartuccelli, M., Constantin, P., Doering, C., Gibbon, J. and Gisselfalt, M., “On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation”, Physica 44D (1990) 421–444.Google Scholar
[3]Brezis, H. and Gallouet, T., “Nonlinear Schrödinger equations”, Nonlinear Anal. 4 (1980) 677–681.CrossRefGoogle Scholar
[4]Bu, C., “On well-posedness of the forced nonlinear Schrödinger equation”, Applicable Anal. 46 (1992) 219–240.CrossRefGoogle Scholar
[5]Bu, C., “The Ginzburg-Landau equation: Posed in a quarter plane”, J. Math. Anal. Appl. 176 (1993) 493–520.CrossRefGoogle Scholar
[6]Carroll, R., Abstract methods in partial differential equations (Harper and Row, New York, 1969).Google Scholar
[7]Carroll, R., “On the forced nonlinear Schrödinger equation”, Japan J. Appl. Math. 7 (1990) 321–344.CrossRefGoogle Scholar
[8]Carroll, R. and Bu, C., “Solutions of the forced nonlinear Schrödinger equation (NLS) using PDE techniques”, Applicable Anal. 41 (1991) 33–51.CrossRefGoogle Scholar
[9]Duan, J. and Holmes, P., “On the Cauchy problem for a generalized Ginzburg-Landau equation”, preprint, 1992.CrossRefGoogle Scholar
[10]Duan, J., Holmes, P. and Titi, E., “Regularity approximation and asymptotic dynamics for a generalized Ginzburg-Landau equation”, Nonlinearity 5 (1992) 1303–1314.CrossRefGoogle Scholar
[11]Fokas, A., “An initial-boundary value problem for the nonlinear Schrödinger equation”, Physica 35D (1989) 167–185.Google Scholar
[12]Ghidaglia, J. and Heron, R., “Dimensions of attractors associated to the Ginzburg-Landau equation”, Physica 28D (1987) 282–304.Google Scholar
[13]Hocking, L., Stewartson, K. and Stuart, J., “A nonlinear instability burst in plane parallel flow”, J. Fluid Mech. 51 (1972).CrossRefGoogle Scholar
[14]Kaup, D., Wave Phenomena (North-Holland, 1984).Google Scholar
[15]Lange, C. and Newell, A., “A stability crireion for the envelope equations”, SIAM J. Appl. Math. 27 (1974) 441–56.CrossRefGoogle Scholar
[16]Lions, J., Quelques méthodes de résolution des problémes aux limites non linéaires (Dunod, Paris, 1969).Google Scholar
[17]Newell, A. and Whitehead, J., “Finite bandwidth, finite amplitude convection”, J. Fluid Mech. 38 (1969).CrossRefGoogle Scholar
[18]Newton, P., “Wave interactions in the singular Zakhorov systems”, J. Math. Phys. 29 (1988) 2245–2249.CrossRefGoogle Scholar
[19]Nirenberg, L., On elliptic partial differential equations (Pisa, 1959) 115–162.Google Scholar
[20]Pazy, A., Semigroups of linear operators and applications to PDE (Springer, New York, 1983).Google Scholar
[21]Segal, I., “Nonlinear semigroups”, Ann. Math. 78 (1963) 339–364.CrossRefGoogle Scholar
[22]Stewartson, K. and Stuart, J., “A non-linear instability theory for a wave system in plane Poiseville flow”,J. Fluid Mech. 48 (1971).CrossRefGoogle Scholar
[23]Strauss, W., “Nonlinear scattering theory at low energy”, J. Funct. Anal. 41 (1981) 110–132.CrossRefGoogle Scholar
[24]Strauss, W., Nonlinear wave equations (CBMS, American Mathematical Society, Rhode Island, 1989).Google Scholar
[25]Tsutsumi, M., “On smooth solutions to the initial-boundary value problem for the nonlinear Schrödinger equation”, Nonlinear Anal. 13 (1989) 1051–1056.CrossRefGoogle Scholar
[26]Tsutsumi, Y., “Global solutions to the nonlinear Schrödinger equation”, Comm. Partial Differential Equations 8 (1983) 1337–1374.CrossRefGoogle Scholar