On a Nonstandard Perturbation Method for Proving the Existence of Nonlinearizable Solutions in a Nonlinear Eigenvalue Problem Arising in Waveguide Theory

The problem of electromagnetic wave propagation in a planar dielectric waveguide is studied. The waveguide is filled with a nonlinear inhomogeneous medium; the nonlinearity is characterized by an arbitrary monotonic positive continuously differentiable function with power-law growth at infinity. The heterogeneity of the medium is characterized by small (nonmonotonic) perturbations of the linear part of the permittivity, as well as the coefficient of the nonlinear term. From a mathematical point of view, this problem is equivalent to a nonlinear eigenvalue problem for the system of Maxwell’s equations with mixed boundary conditions. To study the problem, a perturbation method is proposed with a simpler nonlinear problem as an unperturbed problem. The existence of both linearizable and nonlinearizable solutions is proved.