Propagation of electromagnetic waves in an open planar dielectric waveguide filled with an nonlinear medium I: TE waves

A nonlinear Sturm–Liouville-type eigenvalue problem on an interval with the boundary conditions of the third kind and an additional (local) condition at one of the ends of the interval is studied. The nonlinearity in the equation is specified by a nonnegative monotonically increasing function, and the boundary conditions depend nonlinearly on the spectral parameter. Such a problem describes the propagation of monochromatic electromagnetic TE waves in a planar dielectric waveguide filled with a nonlinear medium. The nonlinearity function covers a wide range of laws of nonlinear optics corresponding to the self-action effects. Results on the solvability of the problem and the properties of eigenvalues are obtained.