On the infinitely many nonperturbative solutions in a transmission eigenvalue problem for Maxwell’s equations with cubic nonlinearity

Abstract

The paper focuses on a transmission eigenvalue problem for Maxwell’s equations with cubic nonlinearity that describes the propagation of transverse magnetic waves along the boundaries of a dielectric layer filled with nonlinear (Kerr) medium. Using an original approach, it is proved that even for small values of the nonlinearity coefficient, the nonlinear problem has infinitely many nonperturbative solutions (eigenvalues and eigenwaves), whereas the corresponding linear problem always has a finite number of solutions. This fact implies the theoretical existence of a novel type of eigenwaves that do not reduce to the linear ones in the limit in which the nonlinear coefficient reduces to zero. Asymptotic distribution of the eigenvalues is found, periodicity of the eigenfunctions is proved, the exact formula for the period is found, and the zeros of the eigenfunctions are determined.