GAMM 2001, Abstract 1

Fedorova A.N., Zeitlin M.G.

LOCALISATION AND COHERENT STRUCTURES IN WAVE DYNAMICS VIA MULTIRESOLUTION

We consider applications of nonlinear (local) harmonic analysis or wavelet analysis to a number of wave motion problems (nonlinear wave equations, Navier-Stokes equation, shock waves, shallow water equation) and turbulence model problem (Kuramoto- Sivashinsky equation). We also consider a number of dynamical problems which are described by the systems of ordinary differential equations with polynomial/rational nonlinearities and with or without some constraints,which are obtained by some anzatzes from preceding wave models. We consider different variational-- wavelet approaches for constructing wavelet representations in multiresolution framework via reduction from initial dynamical problems described by partial/ordinary differential equations to a number of standard algebraical problems. As a result we obtained explicit representation for well localized coherent structures via exact nonlinear multiscale (eigen)mode expansions. The localization is the best both in initial space and in corresponding Fourier space representation. We can compare the contribution to the energy spectrum from each level of resolution or from each scale of full underlying multiscale expansions.