Wavelet Approach to Nonlinear Problems, I. Polynomial Dynamics

this paper we consider the application of powerful methods of wavelet analysis to polynomial approximations of mechanical and physical problems such that accelerator physics problems, dynamics and optimal control problems, Galerkin approximations, routes to chaos in Hamiltonian systems [1]-[7]. The key point in the solution of these problems is the use of the methods of wavelet analysis, relatively novel set of mathematical methods, which gives us a possibility to work with well-localized bases in functional spaces and with the general type of operators (including pseudodifferential) in such bases. Our problem as many related problems in the framework of our type of approximations of complicated nonlinearities is reduced to the problem of the solving of the systems of differential equations with polynomial nonlinearities with or without some constraints. The first main part of our construction is some variational approach to this problem, which reduces initial problem to the problem of solution of functional equations at the first stage and some algebraical problems at the second stage. We consider also two cases of our general construction. In the first case (particular) we have for Riccati equations (particular quadratic approximations) the solution as a series on shifted Legendre polynomials, which is parameterized by the solution of reduced algebraical (also Riccati) system of equations [6]. In the second case (general polynomialsystem) we have the solution in a compactly supported wavelet basis. Multiresolution expansion is the second main part of our construction. The solution is parameterized by solutions of two reduced algebraical problems, one as in the first case and the second is some linear problem, which is obtained from one of the next wavelet construction..