Wave energy flow conservation for propagation in inhomogeneous Vlasov--Maxwell equilibria

Wave energy flow conservation is demonstrated for Hermitian differential operators that arise in the Vlasov--Maxwell theory for propagation perpendicular to a magnetic field. The energy flow can be related to the bilinear concomitant, for a solution and its complex conjugate, by using the Lagrange identity of the operator. This bilinear form obeys a conservation law and is shown to describe the usual Wentzel--Kramers--Brillouin (WKB) energy flow for asymptotically homogeneous regions. The additivity and lack of uniqueness of the energy flow expression is discussed for a general superposition of waves with real and complex wave-numbers. Furthermore, a global energy conservation theorem is demonstrated for an inhomogeneity in one dimension and generalized reflection and transmission coefficients are thereby obtained