Modulated electromagnetic fields in inhomogeneous media, hyperbolic pseudoanalytic functions and transmutations

The time-dependent Maxwell system describing electromagnetic wave propagation in inhomogeneous isotropic media in the one-dimensional case reduces to a Vekua-type equation for bicomplex-valued functions of a hyperbolic variable (see arXiv:1001.0552). Using this relation we solve the problem of the transmission through an inhomogeneous layer of a normally incident electromagnetic time-dependent plane wave. The solution is written in terms of a pair of Darboux-associated transmutation operators (see arXiv:1111.4449), and combined with the recent results on their construction (see arXiv:1208.6166, arXiv:1306.2914) can be used for efficient computation of the transmitted modulated signals. We develop the corresponding numerical method and illustrate its performance with examples.