FULLY NONLINEAR GLOBAL MODES IN SPATIALLY DEVELOPING MEDIA

Global modes on a doubly infinite one-dimensional domain -infinity < x < +infinity are studied in the context of the complex Ginzburg-Landau equation with slowly spatially varying coefficients, i.e., coefficien ts depending only on a slow streamwise coordinate X = epsilon x, where epsilon is a small parameter. A fully nonlinear frequency selection c riterion is derived for global mode solutions under the assumption of weak inhomogeneity of the medium. The global mode is found to be gover ned by the fully nonlinear equations in a region of finite size, and b y the linearized equations in the vicinity of X = +/-infinity. Asympto tic matching techniques are used to relate the WKBJ approximations in the linear and nonlinear regions through appropriate transition layers . The real global frequency is determined by requiring that spatial br anches issuing from X = -infinity and X = +infinity be continuously co nnected at a saddle point of the local nonlinear dispersion relation o mega = Omega(nl)(k, R(2), X) between the frequency omega, the wave num ber k and amplitude R at a given station X. The results constitute a f ully nonlinear generalization of the linear frequency selection criter ia previously obtained by Chomaz et al. (1991), Monkewitz et al. (1993 ), and Le Dizes et al. (1996).