Nonlinear wave propagation in collisionless finite-beta inhomogeneous plasmas

Abstract

Four coupled nonlinear evolution equations for the electrostatic potential, the density fluctuation, and the two vector potentials are derived from the two-fluid and Maxwell’s equations that describe low-frequency collisionless finite-beta inhomogeneous plasmas. Under the assumption of weak turbulence, the above equations reduce to the nonlinear Schrödinger equation. The technique adopted here is considered as an extension of the formal Karpman–Krushkal [Sov. Phys. JETP 28, 277 (1969)] method to a system of nonlinear partial differential equations. The general exact traveling wave solution to the nonlinear Schrödinger equation is obtained with the help of the Hamilton–Jacobi theory. This general solution may be regarded as describing a final nonlinear stage of the modulational instability. It is also shown that a solitary wave solution to the nonlinear Schrödinger equation, which corresponds to the limiting case of the general solution, is given by means of the simple iterative method.