RISC Forum

Special functions and Series appearing in Nonlinear Resonance Analysis

Lena Kartashova

Abstract

Nonlinear Resonance Analysis (NRA) is a natural next step after Fourier analysis developed for linear PDEs. The main subject of NRA is nonlinear PDEs, possessing resonant solutions. The very special role of resonant solutions has been first demonstrated by Poincare, for nonlinear ODEs, and generalized to nonlinear PDEs in the frame of Kolmogorov-Arnold-Moser-theory (KAM-theory). One of the most important and still open questions of NRA is constructing of an appropriate set of simple enough basis functions, similar to Fourier harmonics for linear PDEs.
In this talk I will present my recent results on the topic and give examples of possible prototypes for the set of basis functions, in the form of integrals and/or series depending on Jacobi elliptic functions, elliptic integrals, etc. I would be very grateful for any suggestions on the simplification of the presented formulae!

Importance of NRA is due to its wide application area -- from climate predictability to cancer diagnostic to breaking of the wing of an aircraft.  NRA can also be regarded as a necessary preliminary step for numerical simulations with a big number N of eigen-modes in Galerkin-like methods.  NRA allows reducing drastically the number of eigen-modes (and, correspondingly, the computation time). Namely,

N ~ 10^6 can be reduced to mostly ~10^3,

N ~ 10^3 can be reduced to mostly ~10^2,

N ~ 10 can be reduced to mostly ~10 (in these cases, analytical solutions are often possible to construct and numerical simulations might become superfluous).