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 lled 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 nite number of solutions (eigenvalues and eigenwaves). If a > 0, b 0, then the nonlinear problem has in nitely many solutions; only a nite 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 nite 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.

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

“…The discussed effect is readily observed in similar problems with $$\varepsilon (x) \equiv \mathrm{const}$$ (see Ref. 11).…”

“…We use condition () due to the fact that it is in agreement with physical treatment of problem $$\mathcal {Q}$$ (see Section 7 and also Refs. 9, 11, 12). Using the additional condition in another form can essentially influence on the structure of the set of eigenvalues.…”

“…ICE () is the main to tool to study solvability of problem $$\mathcal {Q}$$. The approach based on usage ICE has already allowed one to solve several complicated problems in this field 9,11,12 …”

“…We use condition (4) due to the fact that it is in agreement with physical treatment of problem  (see Section 7 and also Refs. 9,11,12).…”

“…Previously, the ICE method was suggested for problems of transverse electric and transverse magnetic wave propagation in plane waveguide filled with homogeneous nonlinear medium for a wide range of nonlinearities, [9][10][11] where one has no opportunity to use exact solutions. However, in these cases the derivation of ICE is essentially based on the fact that the governing equations are autonomous.…”