The method of geometrical optics for differential equations of the fourth order as applied to low-frequency plasma oscillations

Abstract

The theory of osculations of a spatially inhomogeneous plasma [1] draws substantially on the theory of geometrical optics as applied to differential equations of the second order. The theory of asymptotic solutions for equations of the second order has now been thoroughly developed [2]. The quasi-classical quantization rules determining the spectrum of eigenvalues of such equations are written in the form of the well-known Bohr -Sommerfeld integrals [3]. However, in analyzing the spectrum of oscillations of an inhomogeneous plasma it is insufficient in many cases to confine oneself to equations of the second order. For example, in an inhomogeneous magnetoactive plasma, even when the thermal motion of the particles is neglected, the field equations, generally speaking, reduce to a differential equation of the fourth order. Equations of the fourth order also arise in investigating the stability of the hydrodynamical flow of a viscous fluid [4].

Certain special forms of fourth-order equations were studied in [4–6]. The authors of [6] obtained a quasi-classical quantization rule for equations of the fourth order with a small parameter associated with the leading derivative. The present paper investigates the general fourth-order equation with real coefficients. Asymptotic solutions of such an equation are obtained with an accuracy to terms of the first order in the approximation of geometrical optics, and quasi-classical quantization rules are established for various concrete cases. Using the theory thus developed, a new spectrum of oscillations is determined, characteristic only for an inhomogeneous plasma in a magnetic field.