Nonlinear propagation of coupled electromagnetic waves in a circular cylindrical waveguide

The problem of the propagation of coupled surface electromagnetic waves in a two-layer cylindrical circular waveguide filled with an inhomogeneous nonlinear medium is considered. A nonlinear coupled TE-TM wave is characterized by two (independent) frequencies $\omega_e$ and $\omega_m$ and two propagation constants $\widehat\gamma_e$ and $\widehat\gamma_m$. The physical problem reduces to a nonlinear two-parameter eigenvalue problem for a system of nonlinear ordinary differential equations. The existence of eigenvalues $(\widehat\gamma_e, \widehat\gamma_m)$ in proven and intervals of their localization are determined.