Symmetric Daubechies' wavelets and numerical solution of NLS equations

Recent work has shown that wavelet-based numerical schemes are at least as effective and accurate as standard methods and may allow an 'easy' implementation of a spacetime adaptive grid. Up to now, wavelets which have been used for such studies are the 'classical' ones (real Daubechies' wavelets, splines, Shannon and Meyer wavelets, etc) and were applied to diffusion-type equations. The present work differs in two points. Firstly, for the first time we use a new set of complex symmetric wavelets which have been found recently. The advantage of this set is 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 equations. The most common method to simulate these models numerically is the symmetrized split-step Fourier method. For the first time, we propose and study a new way of implementing a global spacetime adaptive discretization in this numerical scheme, based on the interpolation properties of complex-symmetric scaling functions. Second, we propose a locally adaptive 'split-step wavelet' method.