Short-wavelength instability in systems with slow long-wavelength dynamics

One-dimensional systems undergoing short-wavelength instability of spatially uniform states are studied. It is assumed that the spectrum of perturbations of the uniform states ${\gamma }_{\mathit{k}}$ has a long-wavelength slowly-relaxing branch, detaching from a neutrally stable (Goldstone) mode with zero wave number, whose existence is a consequence of the problem’s symmetry. The other important feature of the problem is quadratic nonlinearity that provides coupling between slowly-varying short-wavelength and long-wavelength modes. It is shown that the case is characterized by mixing of different scales in perturbative calculations. The latter makes the pattern stability problem essentially nonlocal and sensitive to very subtle characteristics of the spectrum ${\gamma }_{\mathit{k}}$ and nonlinear mode-coupling. The equation governing longitudinal seismic waves in viscoelastic media is studied in detail as the simplest particular example of such systems. Possible extension of the obtained results to other physical problems, including electroconvection in a homeotropically aligned nematic layer and permeation of cholesterics or smectics in capillaries, is discussed. © 1996 The American Physical Society.