WAVELET PROJECTIONS OF THE KURAMOTO-SIVASHINSKY EQUATION .1. HETEROCLINIC CYCLES AND MODULATED TRAVELING WAVES FOR SHORT SYSTEMS

We study fairly low dimensional projections of the Kuramoto-Sivashinsky partial differential equation, with periodic boundary conditions on a short interval, onto bases spanned by periodic wavelets. Such projections break the translation-reflection symmetry (O(2)), replacing it by a finite dihedrai group D-k. However, we show that, for the Perrier-Basdevant wavelets used here, the loss of symmetry is sufficiently mild that key global features of the dynamics are preserved. In particular, we observe heteroclinic cycles and modulated travelling waves arising from interactions of unstable modes on a four dimensional subspace spanned by appropriate combinations of wavelets. We use invariant manifold reductions in our analysis and pay particular attention to symmetries and the relation between periodic wavelets and Fourier modes, which preserve full (O(2)) symmetry and are also optimal in that Fourier truncations maximise the energy (L(2) norm) among all finite dimensional models. This study provides a foundation for current and future work in which we use wavelet bases to extract local models of evolution equations in large space domains.