Nonlinearizable solutions in an eigenvalue problem for Maxwell's equations with nonhomogeneous nonlinear permittivity in a layer

The paper focuses on a problem arising in nonlinear guided wave optics. The problem for Maxwell's equations with nonhomogeneous nonlinear constitutive law is reduced to an eigenvalue problem for a system of nonlinear ordinary differential equations with local boundary conditions. We suggest a novel approach to study eigenvalue problems for nonlinear nonautonomous equations based on studying an integral characteristic equation (ICE) of the problem. Using asymptotical analysis of the ICE, we prove existence of infinitely many eigenvalues and find their distribution. It is important that the corresponding linear problem has only a finite number of eigenvalues.