Kuramoto - Sivashinsky weak turbulence , in the symmetry unrestricted space

Kuramoto-Sivashinsky equation was introduced by Kuramoto [1976] in one-spatial dimension, for the study of phase turbulance in the BelousovZhabotinsky reaction. Sivashinsky derived it independently in the context of small thermal diffusive instabilities for laminar flame fronts. It and related equations have also been used to model directional solidification and , in multiple spatial dimensions, weak fluid turbulence1.2 In recent years unstable periodic orbits have been shown to be an effective tool in the nonlinear dynamical systems3.4The theory has been successfully applied to low-dimensional ordinary differential equations (deterministic chaos) and linear partial differential equations (semiclassical quantization). Since5,6 it has been demonstrated that K-S Equations are rigorously equivalent to a finite dimensional dynamical system of ordinary differential equations(ODE’s). In ref.,7 it was shown that the periodic orbit theory can be used to describe spatially extended systems, by applying it as an example to the Kuramoto-Sivashinsky equation: