Multiple TE-modes for planar, self-focusing wave guides (Q1817910)

The propagation of electro-magnetic TE-waves in planar, multilayered, self-focusing waveguides is studied in detail. It is assumed that the refractive index depends on the intensity of the electric field. The most part of the large article is devoted to the eigenvalues problem \[ Lu+f(u)u=\lambda u, u\in H^1(\mathbb{R})\setminus \{0\}, \] where \(L\) is a linear second order differential operator and \(f(u)\) is a nonlinear term. This problem is solved by variational methods, its solutions can be defined as critical points of some functional \(J(u)\) under the suitable constraints. The critical levels for \(J(u)\) are found, when the critical points have exactly the given number of zeros. The existence of multiple TE-modes is shown in diverse cases and their asymptotic properties are considered.