Modified Nonlinear Schrödinger Equation for Alfvén Waves Propagating along the Magnetic Field in Cold Plasmas

The basic nonlinear equation which describes the Alfvén waves, with small but finite amplitude propagating along the magnetic field in cold plasmas, is derived modifying the reductive perturbation method proposed by Taniuti and Wei. Then as a result, the nonlinear dispersion relation is obtained through a procedure which clarifies the physical meaning. Furthermore, the modified nonlinear Schrödinger equation which describes the modulated Alfvén wave more correctly than the previous works is proposed. An example of the nonlinear phenomena is shown by the numerical calculations of the initial value problem, using our basic equation for the Alfvén waves.