When the Hall effect is included in the magnetohydrodynamics equations (Hall-MHD model) the wave propagation modes become coupled, but for propagation parallel to the ambient magnetic field the Alfvén mode decouples from the magnetosonic ones, resulting in circularly polarized waves that are described by the derivative nonlinear Schr？dinger (DNLS) equation. In this paper, the DNLS equation is numerically solved using spectral methods for the spatial derivatives and a fourth order Runge-Kutta scheme for time integration. Firstly, the nondiffusive DNLS equation is considered to test the validity of the method by verifying the analytical condition of modulational stability. Later, diffusive and excitatory effects are incorporated to compare the numerical results with those obtained by a three-wave truncation model. The results show that different types of attractors can exist depending on the diffusion level: for relatively large damping, there are fixed points for which the truncation model is a good approximation; for low damping, chaotic solutions appear and the three-wave truncation model fails due to the emergence of new nonnegligible modes. 1. Introduction Alfvén waves are one of the most characteristic features of magnetized laboratory and space plasmas. They are driven by different sources, for example, nonuniform plasma parameters, beams of charged particles, and electrostatic and electromagnetic waves [1]. In space plasmas, large MHD amplitude fluctuations with typical proton cyclotron local frequencies were detected on the Earth magnetosphere [2]. Also, nonlinear Alfvén waves were extensively detected in the solar wind [3] and they are believed to be responsible for the turbulent heating of stellar coronas [4]. The comprehension of nonlinear properties of dispersive Alfvén waves is of crucial importance to interpret the abundant amount of low frequency data provided by space plasma observations. On the other hand, the interaction of spatial tethers with the Earth ionosphere and the ambient magnetic field leads to the emission of Alfvén waves forming structures called Alfvén wings. This phenomenon could be applied to produce electric power, generate artificial auroras [5], or to moderate spatial trash [6]. Far from the tethers, a linear analysis could be appropriate [7], but near the conductor intense waves with important nonlinear effects are expected. For the study of the plasma behavior the magnetohydrodynamics (MHD) equations are usually used, but when the frequencies of interest are of the order of the ion-cyclotron frequency or when the
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