Asymptotic analysis of a nonlinear eigenvalue problem arising in electromagnetics

Abstract

An eigenvalue problem for Maxwell’s equations with anisotropic cubic nonlinearity is studied. The problem describes propagation of transverse magnetic waves in a dielectric layer filled with (nonlinear) anisotropic Kerr medium. The nonlinearity involves two non-negative parameters a, b that are usually small. In the case a = b = 0 one arrives at a linear problem that has a finite number of solutions (eigenvalues and eigenwaves). If a > 0, b ⩾ 0, then the nonlinear problem has infinitely many solutions; only a finite number of these solutions have linear counterparts. This shows that perturbation theory methods are inapplicable to study the problem in this case. For a = 0, b > 0 the nonlinear problem has a finite number of solutions; in this case each solution has a linear counterpart. Asymptotic distribution of the eigenvalues is found, periodicity of the eigenfunctions is proved and exact formula for the period is found, zeros of the eigenfunctions are determined, and a (nonlinear) eigenvalue comparison theorem is proved. Numerical experiments are presented.