Stability of periodic waves generated by long-wavelength instabilities in isotropic and anisotropic systems

Abstract

We consider spontaneous generation of long waves in the presence of a conservation law in both cases of isotropic systems (e.g., Bénard–Marangoni waves) and anisotropic systems (e.g., waves in a film on an inclined plane). We found that near the instability threshold the problem is governed by the dissipation-modified Kadomtsev–Petviashvili equation in the former case and by the anisotropic dissipation-modified Korteweg–de Vries equation in the latter case. In frames of the derived 2+1-dimensional amplitude equations, we investigate the stability of one-dimensional waves. In isotropic systems the one-dimensional waves turned out to be always unstable with respect to a long-wave transverse modulation of the front. In anisotropic systems, only the one-dimensional periodic waves moving in the most preferred direction are found to be stable. Any deviation from this direction leads to instability of such an oblique wave.