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{\Large\bf Wavelet-Based Calculation Methods in Nonlinear Collective Dynamics}\
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{\large\bf Antonina Fedorova, Michael Zeitlin}\\
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Institute of Problems of Mechanical Engineering\\
 (IPME RAS) Russian Academy of Sciences\\
Russia, 199178, St. Petersburg, V.O., Bolshoj pr., 61\\
E-mail : anton@math.ipme.ru, zeitlin@math.ipme.ru\\
http://www.ipme.nw.ru/zeitlin.html\\
http://www.ipme.ru/zeitlin.html\\

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{\large\bf Abstract}
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We consider applications of numerical--analytical technique based on the
methods from local nonlinear harmonic analysis to a few models of
nonlinear
collective dynamics both classical and quantum (details are in the first
abstract submitted):
BBGKY hierarchy and general kinetic/master equations,
Vlasov-Maxwell-Poisson based models in beam/plasma
physics, Bose-Einstein condensation [1]-[3].
Our approach is based on a set of methods,
which give the possibility to work with well-localized bases
and provide maximum sparse forms for the general
type of operators in such
bases, and as consequence leads to fast numerical algorithms
in scalar/parallel calculations.
We use fast convergent
variational-wavelet representations for solutions (after Wigner transform
or directly), which allow to
consider
polynomial and rational type of nonlinearities. The solutions are
represented via the multiscale 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. Fast scalar/parallel
modeling demonstrates appearance of high-localized coherent
structures and pattern formation in
spatially--extended stochastic systems with complex collective behaviour [4].

\begin{thebibliography}{4}
\bibitem{1}
A. Fedorova, M. Zeitlin, Localized Coherent Structures and Patterns
Formation in Collective
Models of Beam Motion, arXiv preprint, physics/0101007, World Scientific,
2002, in press
\bibitem{2}
A. Fedorova, M. Zeitlin, Coherent Structures and Pattern Formation in
Vlasov-Maxwell-Poisson
Systems, Proceedings PAC2001, pp.1808-1810, IEEE, 2002,
http://accelconf.web.cern.ch/AccelConf/p01/PAPERS/TPPH064.pdf
or arXiv preprint: physics/0106007
\bibitem{3}
A. Fedorova, M. Zeitlin, Variational-Wavelet Approach to RMS Envelope
Equations, The Physics of High Brightness Beams,
pp.235-254, World Scientific, 2000,
arXiv preprint, physics/0003095
\bibitem{4}
A. Fedorova, M. Zeitlin, From Localization to Stochastics in BBGKY
Collective Dynamics, arXiv preprint: physics/0206049

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