The paper is concerned with propagation of surface TE waves in a circular nonhomogeneous two-layered dielectric waveguide filled with nonlinear medium. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of the Green function. The existence of propagating TE waves for chosen nonlinearity (the Kerr law) is proved using the contraction mapping method. Conditions under which k waves can propagate are obtained, and intervals of localization of the corresponding propagation constants are found. For numerical solution of the problem, a method based on solving an auxiliary Cauchy problem (the shooting method) is proposed. In numerical experiment, two types of nonlinearities are considered and compared: the Kerr nonlinearity and nonlinearity with saturation. New propagation regime is found. 1. Introduction Theory of circle cylindrical dielectric waveguides attracts attention for a long time. Linear theory of such waveguides is known for years; see, for example, [1–4]. At the same time it is well known that the permittivity of a dielectric, in general, depends nonlinearly on the intensity of an electromagnetic field; see [5, 6]. For this reason, the linear theory can be applied only for fields of low intensity. However, for applications, sometimes it is necessary to raise the intensity of the field, for example, in order to compensate the losses. What happens in the case of high intensity (when the permittivity of the dielectric depends nonlinearly on the intensity of the field)? Is it possible to preserve waveguide regimes and, if so, how to determine propagation constants in the nonlinear case? It is not always easy to answer these questions. However, in the case of “simple” geometry (circle cylindrical and plane-layered waveguides) and polarized (TE and TM) electromagnetic waves, it is possible to answer these questions. To the best of our knowledge, the first rigorous study of polarized electromagnetic wave propagation in a nonlinear circle cylindrical dielectric homogeneous waveguide is in [7, 8]. Then, there were several works, where some important cases for nonlinear but homogeneous permittivity have been investigated; see [9–12]. The next step was to apply earlier developed technique to the cases of multilayered waveguides and inhomogeneous nonlinear permittivity. In the paper [13], we considered integral equation approach to derive dispersion equations in a nonlinear waveguiding problem. This approach can be used for numerical implementation. However, there exists a faster and simpler numerical approach that
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