SYMMETRICAL DAUBECHIES WAVELETS AND NUMERICAL-SOLUTION OF NLS EQUATIONS

Recent work has shown that wavelet-based numerical schemes are at leas t as effective and accurate as standard methods and may allow an 'easy ' implementation of a spacetime adaptive grid. Up to now, wavelets whi ch have been used for such studies are the 'classical' ones (real Daub echies' wavelets, splines, Shannon and Meyer wavelets, etc) and were a pplied to diffusion-type equations. The present work differs in two po ints. Firstly, for the first time we use a new set of complex symmetri c wavelets which have been found recently. The advantage of this set i s that, unlike classical wavelets, they are simultaneously orthogonal, compactly supported and symmetric. Secondly, we apply these wavelets to the physically meaningful cubic and quintic nonlinear Schrodinger e quations. The most common method to simulate these models numerically is the symmetrized split-step Fourier method. For the first time, we p ropose and study a new way of implementing a global spacetime adaptive discretization in this numerical scheme, based on the interpolation p roperties of complex-symmetric scaling functions. Second, we propose a locally adaptive 'split-step wavelet' method.