Stability of steady spatially periodic patterns in systems with an additional continuous group of symmetry is discussed. It is shown that different systems with the same dimensionality of the continuous group of symmetry display remarkable similarity in all qualitative features of the pattern stability problem. Attention is called to the fact that, beside an extra band of slowly varying modes, the additional symmetry may yield a mixture of different scales in the final dispersion equation for pattern's perturbations, so that the stability conditions become unusually sensitive to very fine details of the problem. A one-dimensional partial differential equation governing seismic waves in viscoelastic media is considered as a particular example. The equation exhibits short-wave instability and additional invariance under the transformation u → u + const. , where the order parameter u(x, t) is associated with the displacement velocity. The analytical study of the equation is supplemented by computer simulations. The simulations show that the system undergoes a bifurcation from the trivial state with u ≡ 0 to well-developed chaos directly and the transition to the chaos is smooth, without any discontinuity. The chaos is characterized by excitation of a big number (in a boundless system — continuum) of coupled modes localized generally in two narrow subbands, centered around the critical wavenumber for the short-wave instability and the wavenumber equals zero, respectively.
Short wave instability of co-extruded elastic liquids with matched viscosities
