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Self-trapping of an electromagnetic field and bifurcation from the essential spectrum. (English) Zbl 0745.35044

(From author’s summary.) This paper has two main aims: (i) the representation of a mathematical description of a problem concerning the self-trapping of an electromagnetic wave, and (ii) the discussion of the relationship between this problem and the phenomenon of bifurcation from the essential spectrum. In the first part, we recall the basic physical quantities that are required and then we show how the problem can be reduced to a convenient mathematical form. In contrast to the approximate equations usually used in discussing self-trapping the solutions of our reduced problem yield exact solutions of Maxwell’s equations within the context of the constitutive hypotheses generally adopted in the optical regime with an intensity-dependent refractive index.
In the second part, the reduced problem is discussed from a variational point of view. By avoiding the usual approximations, we can clearly identify the precise physical significance of the parameters and norms used in this analysis. In particular, we observe that bifurcation from the essential spectrum with respect to the \(L^ 2\)-norm is equivalent to the occurrence of self-trapping at arbitrarily small intensities.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
35P05 General topics in linear spectral theory for PDEs
78A25 Electromagnetic theory (general)
35A15 Variational methods applied to PDEs
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