Numerical Study of Attractors for the Diffusive DNLS Equation using Spectral Methods

摘要

The Derivative Nonlinear Schrodinger (DNLS) equation arises from the magnetohydrodynamics (MHD) model for parallel to the ambient magnetic field propagation when the Hall term is preserved. In that case, the Alfven mode deacouples from the magnetosonic modes and the waves are circularly polarized being described by the DNLS equation. In this work, the diffusive DNLS equation for periodic boundary condition is numerically solved using spectral methods for the spatial derivatives and a fourth order Runge-Kutta scheme for the time integration. The aim is to study attractors numerically obtained when a three-mode initial condition is used with one mode excited and the others damped under a resonant relation. In order to know how the energy transfer is produced, the results are compared with a three-mode truncation model. It is shown that for relatively small diffusion values, the three-mode model fails because the energy is transfered to two additional modes, which are related to the initial ones by a new resonance relation.