Invariant imbedding approach to localization. I. General framework and basic equations

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Abstract
A unified framework for solving the problems of wave transmission across a random medium is outlined. Using an invariant imbedding approach, differential equations are derived for the reflection and transmission coefficients. In general, the transmission problem, viewed as a boundary-value problem, can be reduced to an initial-value Cauchy equation, relative to the imbedding parameters. Known results are recovered and new equations pertaining to multichannel problems, time-dependent medium, etc., are obtained. The extension of this approach to other cases is outlined. A systematic method for the investigation of the stochastic differential equations so obtained is described. The case of one-dimensional linear media is used as an illustrative example