A plane multilayered waveguide structure is considered. The layers are located between two half-spaces with constant permittivities. The permittivity inside each layer depends on modulus of the electric field intensity by arbitrary law. This structure can be treated as a 1D (nonlinear) photonic crystal. We consider propagation of polarized electromagnetic waves in such a structure. Surface waves are sought for. The physical problem is to determine propagation constants of electromagnetic waves propagating in the waveguide. This problem is reduced to (nonlinear) conjugation eigenvalue problem in a multiply-connected domain. Usually, in such problems, the main goal is to obtain a dispersion equation (DE) for propagation constants (eigenvalues). For many physically interesting nonlinear permittivities it is far beyond our abilities to obtain and analyze exact DEs. We suggest a numerical approach to calculate propagation constants (eigenvalues) for (nonlinear) multilayered waveguide structures based on numerical solution of a Cauchy problem in each layer.
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