Wavelet Analysis As A Tool For Solution Of Nonlinear Problems
by
Michael Zeitlin
Head of Mathematical Methods in Mechanics Group, Institute of Problems of Mechanical Engineering, Russian Academy of Scis
Coauthors: Antonina Fedorova

PART 1. We consider a number of dynamical problems which are described by the systems of ordinary differential equations with polynomial nonlinearities and with or without some constraints (dynamical problems, Galerkin approximations for nonlinear partial differential equations, optimal control problems). We consider variational approach for constructing explicit solutions of these problems. We have the solutions as a multiresolution expansion in the base of compactly supported wavelets or wavelet packet bases. In the general case we consider biorthogonal wavelet expansions. These solutions are parameterized by solutions of reduced algebraical problems. We consider practical applications of our computations to satellites, submarines and related problems.

PART 2. We consider applications of methods from wavelet analysis to a number of (optimal) control (electromechanical systems, robots, spacecrafts, satellites) and constrained problems (homoclinic loops). We present constructions which give explicit in time representations for the controlling and controllable variables in different types of well localized bases for the general classes of functional spaces with sparse representation for the corresponding operators. We also consider the problem of practical realization of the computed solutions and applications to problems of chaotic dynamics.

PART 3. We consider the applications of methods from wavelet analysis to a number of wave motion problems (nonlinear wave equations) and turbulence problems (Kuramoto-Sivashinsky equations). We consider variational approach for constructing wavelet-Galerkin representations via reduction from initial complicated problems to a number of standard algebraical problems. As a result we obtained explicit representation for well localized coherent structures. Also we consider Hamiltonian contents of these problems.

PART 4. We present the applications of the methods from wavelet analysis to a number of Hamiltonian problems and their perturbations. We consider dynamical problems via coadjoint orbit picture and metaplectic structure. We construct symplectic, Poisson and quasicomplex structures using generalized wavelets and non-standard (maximum sparse) representations for operators in wavelet bases (coherent, well localized) in functional spaces or scale of spaces. We consider applications of our approach to calculations of Melnikov functions in the theory of homoclinic chaos. We give parametrization of periodic solutions by some wavelet construction.

Date received: March 2, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caav-12.