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-Sivashinsk y partial differential equation, with periodic boundary conditions on a short interval, onto bases spanned by periodic wavelets. Such projec tions break the translation-reflection symmetry (O(2)), replacing it b y a finite dihedrai group D-k. However, we show that, for the Perrier- Basdevant wavelets used here, the loss of symmetry is sufficiently mil d that key global features of the dynamics are preserved. In particula r, we observe heteroclinic cycles and modulated travelling waves arisi ng from interactions of unstable modes on a four dimensional subspace spanned by appropriate combinations of wavelets. We use invariant mani fold reductions in our analysis and pay particular attention to symmet ries and the relation between periodic wavelets and Fourier modes, whi ch preserve full (O(2)) symmetry and are also optimal in that Fourier truncations maximise the energy (L(2) norm) among all finite dimension al models. This study provides a foundation for current and future wor k in which we use wavelet bases to extract local models of evolution e quations in large space domains.