Huygens' Principle in the narrow sense means that an observer would feel a momentary disturbance if the disturbance was initially localised at some point. Such a phenomenon holds true for acoustic or electromagnetic waves in our space making it possible for us to communicate and hear each other, to transmit information. Remarkably the wave equation on the plane does not have this property.
In 1923 J.Hadamard raised a question of how special the wave equation in three-dimensional space is and whether it can be modified still retaining the Huygens' Principle. As he expected it happened that such a modification is impossible in our space, however it can be done in the higher-dimensional spaces. To describe these equations is a challenging open problem.
The Hadamard's problem happens to be strongly related to the theory of integrable systems, in particular to Darboux transformations in many dimensions. Some equations satisfying the Huygens Principle are given in terms of rational and solitonic solutions of the Korteweg - de Vries equation. Another source for these equations are integrable systems of Calogero-Moser type. These systems are characterised by special arrangements of hyperplanes. As an important example one can can take the symmetry planes of the Platonic solids but in general the geometry of these hyperplanes is more puzzling.
- O.A.Chalykh, M.V.Feigin, A.P.Veselov, Multidimensional Baker-Akhiezer Functions and Huygens' Principle, Commun. Math. Phys., 1999, 206, pp. 533-566
- O.A.Chalykh, M.V.Feigin, A.P.Veselov, New integrable generalizations of Calogero-Moser quantum problem, J. Math. Phys., 1998, 39 (2), pp. 695-703