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

We consider spontaneous generation of long waves in the presence of a conse rvation law in both cases of isotropic systems (e.g., Benard-Marangoni wave s) 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 d issipation-modified Kadomtsev-Petviashvili equation in the former case and by the anisotropic dissipation-modified Korteweg-de Vries equation in the l atter case. In frames of the derived 2+1-dimensional amplitude equations, w e 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, onl y the one-dimensional periodic waves moving in the most preferred direction are found to be stable. Any deviation from this direction leads to instabi lity of such an oblique wave. (C) 1999 Elsevier Science B.V. All rights res erved.