No. 30 (00112) Family name : Fedorova Given name : Antonina Affiliation : Mathematical Methods in Mechanics Group, Institute of Problems of Mechanical Engineering, Russian Academy of Sciences Abbreviation : IPME RAS E-mail address : anton@math.ipme.ru Title : LOCALIZATION AND PATTERNS FORMATION IN (NONLINEAR) COLLECTIVE DYNAMICS Authors : Antonina N. Fedorova, Michael G. Zeitlin Abstract : We demonstrate the appearance of nontrivial localized stable states/patterns in a number of well-known collective models such as BEC, nonlinear (quantum) optics models, the (quantum) BBGKY hierarchy of kinetic equations, Vlasov-Maxwell collective models of beam/plasma physics etc and present the explicit constructions for exact analytical/numerical computations. Our fast and efficient approach is based on variational and multiresolution representations in the bases of polynomial tensor algebras of generalized coherent states/wavelets (fast convergent variational-wavelet representation). We construct the representations for hierarchy/algebra of observables/distribution functions via the complete multiscale decompositions, which allow to consider the polynomial and rational type of nonlinearities. The solutions are represented via the exact decomposition in nonlinear high-localized eigenmodes, which correspond to the full multiresolution expansion in all underlying hidden time/space or phase space scales. In contrast with different approaches we do not use perturbation technique or linearization procedures. Numerical modeling shows the creation of different internal structures from localized modes, which are related to the localized or chaotic type of behaviour and the corresponding patterns (waveletons) formation.