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Asymptotic analysis of a nonlinear eigenvalue problem arising in electromagnetics. (English) Zbl 1440.35321

Summary: 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 \(a > 0\), \(b \geqslant 0\), 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.

MSC:

35Q61 Maxwell equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
78A60 Lasers, masers, optical bistability, nonlinear optics
35B40 Asymptotic behavior of solutions to PDEs
78A40 Waves and radiation in optics and electromagnetic theory
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