Damping models in the truncated derivative nonlinear Schrödinger equation

Abstract

Four-dimensional flow in the phase space of three amplitudes of circularly polarized Alfven waves and one relative phase, resulting from a resonant three-wave truncation of the derivative nonlinear Schrödinger equation, has been analyzed; wave 1 is linearly unstable with growth rate , and waves 2 and 3 are stable with damping γ2 and γ3, respectively. The dependence of gross dynamical features on the damping model (as characterized by the relation between damping and wave-vector ratios, γ2 γ3, k2 k3), and the polarization of the waves, is discussed; two damping models, Landau (γk) and resistive (γ k2), are studied in depth. Very complex dynamics, such as multiple blue sky catastrophes and chaotic attractors arising from Feigenbaum sequences, and explosive bifurcations involving Intermittency-I chaos, are shown to be associated with the existence and loss of stability of certain fixed point P of the flow. Independently of the damping model, P may only exist for <2 (γ2 + γ3) 3, as against flow contraction just requiring < γ2 + γ3. In the case of right-hand (RH) polarization, point P may exist for all models other than Landau damping; for the resistive model, P may exist for RH polarization only if < (γ2 + γ3) 2. © 2007 American Institute of Physics.