Solutions of a fourth-order differential equation describing mode conversion in a magnetized plasma

Abstract

Solutions of the equation ε2u(4)+zu(2)+αu(1)+β1u −β22zu=0, where α, β1, β22 are constants, ε2≪1, and z is the independent variable, are obtained using the Laplace integral technique. This equation describes the propagation of high frequency electrostatic waves near plasma resonance in a magnetized plasma with a longitudinal density gradient and is a generalization of an equation studied by Wasow and by Rabenstein in the context of boundary layer phenomena. The solutions of this fourth-order equation in which the associated second-order equation (i.e., ε2=0) exhibits both a singularity (at z=0) and a turning point (at z=β1/β22) fall readily into two classes. One class resembles Airy functions and exists only for ε2 not equal to zero. In the other class, the solutions are related to confluent hypergeometric functions and can be viewed as solutions of the second-order equation with small corrections proportional to ε2. Using the integral representations of solutions, it is demonstrated that each class of solutions can generate the other when the independent variable crosses the singular point. This is the physical phenomenon of mode conversion. Asymptotic descriptions of both classes of solutions are given and the form of the solutions near the singular point is expressed as a power series.