ABSTRACT
The wavelet representation of a time-dependent signal can be used to study the propagation of energy between the different scales in the signal. Burgers’ evolution operator (in 1 and 2 dimensions) can itself be described from this scaling point of view. Using wavelet-based algorithms we can depict the transfer of energy between scales. We can write the instantaneous evolution operator in the wavelet basis; then large off-diagonal terms will correspond to energy transfers between different scales. We can project the solution onto each fixed-scale wavelet subspace and compute the energy; then the rate of change of this energy by scale can detect and quantify any cascades that may be present. These methods improve the classical Fourier-transform-based scale decomposition which uses the notion that wavenumber equals scale. The wavelet basis functions underlying our scale decompositions have finite, well-defined position uncertainty (i.e., scale) whereas Fourier basis functions have formally unbounded position uncertainty.
