Electromagnetic TE wave propagation in an inhomogeneous nonlinear cylindrical waveguide is considered. The permittivity inside the waveguide is described by the Kerr law. Inhomogeneity of the waveguide is modeled by a nonconstant term in the Kerr law. Physical problem is reduced to a nonlinear eigenvalue problem for ordinary differential equations. Existence of propagating waves is proved with the help of fixed point theorem and contracting mapping method. For numerical solution, an iteration method is suggested and its convergence is proved. Existence of eigenvalues of the problem (propagation constants) is proved and their localization is found. Conditions of k waves existence are found. 1. Introduction Electromagnetic wave propagation in linear (homogeneous and inhomogeneous) waveguide plane layers and cylindrical waveguides with circular cross section is of particular interest in linear optics (see, e.g., [1, 2]). In nonlinear optics, waveguides (plane and cylindrical) filled with nonlinear medium have been the focus of a number of studies [3–11]. However, many of researches are devoted to study homogeneous nonlinear waveguides [6–11]. Problems of electromagnetic wave propagation in nonlinear waveguides (plane and cylindrical) lead to nonlinear boundary and transmission eigenvalue problems for ordinary differential equations. Eigenvalues in these problems correspond to propagation constants of the waveguides. In these problems differential equations depend nonlinearly either on sought-for functions and the spectral parameter. Boundary and/or transmission conditions depend nonlinearly on the spectral parameter. The main goal is to prove existence of eigenvalues and determine their localization. Existence and localization can be derived from the dispersion equation (DE). DE is an equation with respect to spectral parameter. There are two ways to obtain the DE. The first one is to integrate the differential equations and obtain, using boundary and/or transmission conditions, the DE. This way is of very limited applicability, as it is very rarely possible to find explicit solutions of nonlinear differential equations. However, there are some problems in which this way works (see, e.g., [10, 12, 13]). The second one is a very general approach based on reduction of the differential equations to integral equations using the Green function. This approach we call integral equation approach. Here we consider this very method. Inspite of the fact that by this method the DE is found in an implicit form, it is possible to prove existence of eigenvalues and find
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