Nonlinear transmission eigenvalue problem describing TE wave propagation in two-layered cylindrical dielectric waveguides

The paper is concerned with propagation of surface TE waves in a circular nonhomogeneous two-layered dielectric waveguide filled with a Kerr nonlinear medium. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of a Green’s function. The existence of propagating TE waves is proved using the contraction mapping method. For the numerical solution of the problem, two methods are proposed: an iterative algorithm (whose convergence is proved) and a method based on solving an auxiliary Cauchy problem (the shooting method). The existence of roots of the dispersion equation (propagation constants of the waveguide) is proved. Conditions under which k waves can propagate are obtained, and regions of localization of the corresponding propagation constants are found.