The stability of standing waves with small group velocity
The subject of the work is a system of two nonlinearly coupled mutually symmetric cubic complex Ginzburg-Landau equations, each having a group-velocity term with opposite signs of the group velocities. This model describes interaction of counterpropagating traveling waves in a layer of binary fluid heated from below, when the layer is close to the onset of the convective instability. A well-known class of solutions to the system are standing waves. An objective of the present work is to study in detail stability of the standing-wave solutions, and thus to identify, for the standing waves, an analog of the so-called Eckhaus stability region, which is a well-known range of stability of traveling (rather than standing) waves in similar models. The analysis reveals two general mechanisms of destabilization of the standing-wave solutions by long-wave perturbations, and also a mechanism generating a short-wave instability. As a result, the Eckhaus parabola bordering the stability region for the standing waves is shifted off-center. The analytical consideration is supplemented by direct simulations of the system. Close to a transition between standing and traveling waves, the simulations reveal very complex chaotic-like dynamics, which may be realized as a result of the competition between the counterpropagating waves in an interplay with the previously known process of wavenumber selection by local solutions in the form of wave sources. Crossover from the considered model to more general nonlocal and higher-order (hyperbolic) models, which asymptotically reduce to the present one close to the onset of the instability of the zero solution, is considered in detail too.
interaction of counterpropagating traveling waves
scientific article; zbMATH DE number 1417475
scientific article