Numerical simulations of the Complex Ginzburg-Landau Equation Linda El Alaoui LAGA Universit´e Paris 13
Hatem Zaag CNRS LAGA Universit´e Paris 13
November 19, 2009 Consider the following Complex Ginzburg-Landau equation ∂t u = (1 + iβ)∆u + (ǫ + iδ)|u|p−1 u − γu,
(1)
where u = u(x, t) ∈ C, x ∈ RN , t ≥ 0, p > 1, ǫ = ±1, β, δ and γ are real constants. This equation appears in the study of various physical problems (plasma physics, nonlinear optics, etc...). When ǫ = 1, Masmoudi and Zaag [2] prove that this equation admits blow-up solutions u(x, t), i.e. solutions which exist for all t ∈ [0, T ) for some T > 0 and satisfy lim ku(t)kL∞ (RN ) = +∞. t→T
This result confirms a formal evidence given by Popp et al [3]. The aim of this project is to make numerical simulations of equation (1), in the blow-up regime ǫ = 1.. Since the equation is invariant under the following scaling (when γ = 0): √ 1 λ 7→ uλ (ξ, τ ) = λ p−1 u( λξ, λτ ), we proceed like Berger and Kohn [1] in the case β = δ = γ = 0 and adopt an iterative numerical method, where we refine the mesh whenever the maximum of the solution achieves a certain large constant M . The project is progressive, in the sense that we first start from the case of Berger and Kohn (β = δ = γ = 0), then we take δ 6= 0 and finally the full model with β and δ not necessarily zero. In the same time, we first handle the one dimensional case (N = 1), then move to higher dimensions N ≥ 2). Contact: Linda El Alaoui:
[email protected] Hatem ZAAG:
[email protected] Duration: 3 months (April - May - June, or contact in case of other dates) Financial conditions: Around 600 euros per month (depending on excellence of the student). 1
References [1] M. Berger and R. V. Kohn. A rescaling algorithm for the numerical calculation of blowing-up solutions. Comm. Pure Appl. Math., 41(6):841–863, 1988. [2] N. Masmoudi and H. Zaag. Blow-up profile for the complex Ginzburg-Landau equation. J. Funct. Anal., 255(7):1613–1666, 2008. [3] S. Popp, O. Stiller, E. Kuznetsov, and L. Kramer. The cubic complex Ginzburg-Landau equation for a backward bifurcation. Phys. D, 114(1-2):81–107, 1998.
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