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.

中文翻译：

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电磁学中非线性特征值问题的渐近分析

研究了具有各向异性三次非线性的麦克斯韦方程组的特征值问题。该问题描述了横向磁波在充满（非线性）各向异性克尔介质的介电层中的传播。非线性涉及两个通常很小的非负参数 a、b。在 a = b = 0 的情况下，一个线性问题得到了有限数量的解（特征值和特征波）。如果a > 0，b ≥ 0，则非线性问题有无穷多个解；只有有限数量的这些解决方案具有线性对应项。这表明微扰理论方法不适用于研究这种情况下的问题。对于 a = 0, b > 0，非线性问题的解数是有限的；在这种情况下，每个解决方案都有一个线性对应项。发现特征值的渐近分布，证明了特征函数的周期性并找到了周期的精确公式，确定了特征函数的零点，并证明了（非线性）特征值比较定理。给出了数值实验。