Bifurcation to fully nonlinear synchronized structures in slowly varying media

The selection of fully nonlinear extended oscillating states is analyzed in the context of one-dimensional nonlinear evolution equations with slowly spatially varying coefficients on a doubly infinite domain. Two types of synchronized structures referred to as steep and soft global modes are shown to exist. Steep global modes are characterized by the presence of a sharp stationary front at a marginally absolutely unstable station and their frequency is determined by the corresponding linear absolute frequency, as in Dee–Langer propagating fronts. Soft global modes exhibit slowly varying amplitude and wave number over the entire domain and their frequency is determined by the application of a saddle point condition to the local nonlinear dispersion relation. The two selection criteria are compared and shown to be mutually exclusive. The onset of global instability first gives rise to a steep global mode via a saddle-node bifurcation as soon as local linear absolute instability is reached somewhere in the medium. As a result, such self-sustained structures may be observed while the medium is still globally stable in a strictly linear approximation. Soft global modes only occur further above global onset and for sufficiently weak advection. The entire bifurcation scenario and state diagram are described in terms of three characteristic control parameters. The complete spatial structure of nonlinear global modes is analytically obtained in the framework of WKBJ approximations.