Stationary nonlinear Alfvén waves and solitons

Stationary solutions of the derivative nonlinear Schroedinger equation are discussed and classified by using a pseudopotential formulation. The solutions consist of a rich family of nonlinear Alfvén waves and solitons with parallel and oblique propagation directions. Expressions for the envelope and the phase of nonlinear waves with periodic envelope modulation, and “hyperbolic” and “algebraic” solitons are given. We evaluate the propagation angle for the slightly modulated elliptic, periodic waves and for oblique solitons. Also, we present periodic stationary waves which may arise in numerical simulations using periodic boundary conditions. The parallel and oblique stationary solutions discussed here can serve as a starting point for studies of modulational and decay instabilities and for a stability analysis of the solitons.