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Huygens’ principle and integrability. (English. Russian original) Zbl 0941.35003
Russ. Math. Surv. 49, No. 6, 5-77 (1994); translation from Usp. Mat. Nauk 49, No. 6, 7-78 (1994).
The authors give a survey on Huygens’ principle (HP) in the sense of Hadamard and Courant/ Hilbert: a linear hyperbolic differential equation or equation system is said to satisfy the HP if the solution at a point \(x\) of a Cauchy problem depends only on the initial data on \(C_x\cap S\), where \(C_x\) denotes the past characteristic conoid with vertex \(x\) with respect to the Lorentzian metric defined by the equation, and \(S\) denotes the initial hypersurface. This means that the solution is a wave which propagates with maximal velocity; there are no aftereffects. Older surveys and, in particular, the book by P. Günther [Huygens’ principle and hyperbolic equations, Academic Press, Boston, MA (1988; Zbl 0655.35003)] presented most of the results on HP known in their times. Meanwhile there has been further progress.
Berest and Veselov’s paper surveys both the history of HP since Hadamard introduced it in 1923 and modern achievements, partly coming from the authors themselves. In Section 1 they present Hadamard’s singularity analysis and note the close relation to the so-called relative Painlevé property. In Section 2 Mathisson’s theorem, “A scalar Huygens-type equation on the four-dimensional flat spacetime is transformable to the ordinary wave equation”, is derived and counterexamples with altered dimension and/or signature of the metric are added to it. Next comes Asgeirsson’s and Günther’s result that HP and the existence of a “diversor” and the existence of a “wave family” are properties equivalent to each other. This is supplemented by a seemingly new relation between Laplace’s cascade method and the Toda chain. Section 3 discusses the Stellmacher-Lagnese class of Huygens-type equations, including their relation to the Korteweg-de Vries hierarchy. Just this class is vastly extended by the authors in Section 4 in a most interesting way with the help of Coxeter groups (i.e. finite reflection groups), Dunkl operators, intertwining relations, and Baker-Akhiezer functions. Section 5 reviews papers of Günther, Ibragimov et al.; here the proper name “plane gravitational wave” for the metric under consideration should have been given. Finally, Section 6 treats the wave equation on a symmetric space, following Helgason et al.
It can be expected that the present paper will become a standard reference on Huygens’ principle.

MSC:
35Axx General topics in partial differential equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35L15 Initial value problems for second-order hyperbolic equations
58J47 Propagation of singularities; initial value problems on manifolds
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