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Maxwell’s equations, the Euler index, and Morse theory. (English. Russian original) Zbl 1358.35183

Math. Notes 100, No. 3, 352-362 (2016); translation from Mat. Zametki 100, No. 3, 331-343 (2016).
Summary: We show that the singularities of the Fresnel surface for Maxwell’s equation on an anisotrpic material can be accounted from purely topological considerations. The importance of these singularities is that they explain the phenomenon of conical refraction predicted by Hamilton. We show how to desingularise the Fresnel surface, which will allow us to use Morse theory to find lower bounds for the number of critical wave velocities inside the material under consideration. Finally, we propose a program to generalise the results obtained to the general case of hyperbolic differential operators on differentiable bundles.

MSC:

35Q61 Maxwell equations
35Q60 PDEs in connection with optics and electromagnetic theory
78A05 Geometric optics
58B05 Homotopy and topological questions for infinite-dimensional manifolds
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
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References:

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