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Renormalization group and the Ginzburg-Landau equation. (English) Zbl 0765.35052
Summary: We use renormalization group methods to prove detailed long time asymptotics for the solutions of the Ginzburg-Landau equations with initial data approaching, as $x\to\pm\infty$, different spiraling stationary solutions. A universal pattern is formed, depending only on this asymptotics at spatial infinity.

35Q55NLS-like (nonlinear Schrödinger) equations
81T17Renormalization group methods (quantum theory)
35B40Asymptotic behavior of solutions of PDE
Full Text: DOI
[1] Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math.30, 33--76 (1978) · Zbl 0407.92014 · doi:10.1016/0001-8708(78)90130-5
[2] Ben-Jacob, E., Brand, H., Dee, G., Kramers, L., Langer, J.S.: Pattern propagation in nonlinear dissipative systems. Physica14D, 348--364 (1985) · Zbl 0622.76051
[3] Bramson, M.: Convergence of solutions of the Kolmogorov equation to traveling waves. Mem. of the Am. Math. Soc.44, nr. 285, 1--190 (1983) · Zbl 0517.60083
[4] Bricmont, J., Kupiainen, A., Lin, G.: Renormalization group and asymptotics of solutions of nonlinear parabolic equations. To appear in Commun. Pure Appl. Math. · Zbl 0806.35067
[5] Bricmont, J., Kupiainen, A.: In preparation
[6] Collet, P., Eckmann, J.-P.: Solutions without phase-slip for the Ginsburg-Landau equation. Commun. Math. Phys. (1992) · Zbl 0748.35044
[7] Collet, P., Eckmann, J.-P.: Space-time behaviour in problems of hydrodynamic type: a case study. Preprint (1992) · Zbl 0757.35059
[8] Collet, P., Eckmann, J.-P., Epstein, H.: Diffusive repair for the Ginsburg-Landau equation. Helv. Phys. Acta65, 56--92 (1992)
[9] Dee, G.: Dynamical properties of propagating front solutions of the amplitude equation. Physica15D, 295--304 (1985) · Zbl 0575.76048
[10] Goldenfeld, N., Martin, O., Oono, Y., Liu, F.: Anomalous dimensions and the renormalization group in a nonlinear diffusion process. Phys. Rev. Lett.64, 1361--1364 (1990) · doi:10.1103/PhysRevLett.64.1361
[11] Goldenfeld, N., Martin, O., Oono, Y.: Asymptotics of partial differential equations and the renormalization group. To appear in the Proc. of the NATO ARW on Asymptotic beyond all orders. Ed. by S. Tanveer, Plenum Press · Zbl 0925.73081