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Benoît PIER and Patrick HUERRE
LadHyX, École polytechnique - CNRS, 91128 Palaiseau Cedex, France
Physica D 97, 206–222 (1996)
Global modes on a doubly-infinite one-dimensional domain $-\infty < X < +\infty$ are studied in the context of the complex Ginzburg-Landau equation with slowly spatially varying coefficients. A fully nonlinear frequency selection criterion is derived for global-mode solutions under the assumption of weak inhomogeneity of the medium. The global mode is found to be governed by the fully nonlinear equations in a region of finite size, and by the linearized equations in the vicinity of $X=\pm\infty$. Asymptotic matching techniques are used to relate the WKB approximations in the linear and nonlinear regions through appropriate transition layers. The real global frequency is determined by requiring that spatial branches issuing from $X=-\infty$ and $X=+\infty$ be continuously connected at a saddle point of the local nonlinear dispersion relation $\omega=\Omega^{nl}(k,R,X)$ between the frequency $\omega$, the wavenumber $k$ and amplitude $R$ at a given station $X$. The results constitute a fully nonlinear generalization of the linear frequency selection criteria previously obtained by Chomaz et al. (1991), Monkewitz et al. (1993), and Le Dizès et al. (1996).
hal-00119915
1996a_pier_physD.pdf