
made
of only 5 modes. In the next simulations, we consider this
particular initial value, with ε
= 0.5.The
numerical scheme used is based on a splitting algorithm
between the linear and nonlinear parts, that both can be
solved exactly in Fourier and (x,y) variables respectively.
This algorithm can be implemented in an efficient way using
the Fast Fourier Transform algorithm. The main issues with
this method are aliasing problems due to the space
discretization, and stepsize resonances due to the time
integration. We refer to the book Geometric
numerical integration and Schrödinger equations, by
E. Faou, European Math. Soc. 2012 for discussions and
convergence proofs of this numerical scheme.
To avoid this numerical resonances problem, we systematically
use stepsizes satisfying a CourantFriedrichsLewy condition
of the form τN^{2} <
c, where τ is
the time step. Henre, N = 32,64,128 and 256 is the number of
modes in each direction.
Let us make some comments: the previous simulations do not contradict the existing bounds of J. Bourgain, W.M. Wang or J.M. Delort, (see for instance JeanMarc's paper here), essentially because the potential we consider has a large derivative with respect to the time (though is is bounded and smooth in the space variable). However, the nonlinear effect of the oscillating potential can be observed in a relatively short time. And it gives nice pictures!