On solving transport equations

Abstract

THE widely known asymptotic method of solving partial differential equations (see [1]) leads to an infinite recurrence system of ordinary differential equations along the bicharacteristics (rays, in physical terminology). Solving the first of these equations is of most interest, since in many practical applications of the asymptotic method the solution is confined to the first term of the expansion. The present note deals with systems of 1st and 2nd order partial differential equations having both distinct and multiple real characteristics. In the latter case it is found that, for every k-tuple characteristic surface, the first transport equation reduces to a system of ordinary differential equations describing the rotation, while travelling along the bicharacteristic, of some k-component vector. Physical analogues of this vector are the electromagnetic field polarization (Maxwell's equations), the spin (Dirac' equations) and the polarization of transverse elastic waves (the equations of the theory of elasticity). Another important circumstance is that, in the cases of either multiple or distinct characteristics, { ∂(x 1, …, x n) (a 1, …, a n) } − 1 2 separates out as a factor in the solution; this is vital when investigating the behaviour of the solution of partial differential equations as a whole [2].