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Hierarchies of Huygens' Operators and Hadamard's Conjecture

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Abstract

We develop a new unified approach to the problem of constructing linear hyperbolic partial differential operators that satisfy Huygens' principle in the sense of J. Hadamard. The underlying method is essentially algebraic and based on a certain nonlinear extension of similarity (gauge) transformations in the ring of analytic differential operators.

The paper provides a systematic and self-consistent review of classical and recent results on Huygens' principle in Minkowski spaces. Most of these results are carried over to more general pseudo-Riemannian spaces with the metric of a plane gravitational wave.

A particular attention is given to various connections of Huygens' principle with integrable systems and the soliton theory. We discuss the link to nonlinear KdV-type evolution equations, Darboux–Bäcklund transformations and the bispectral problem in the sense of Duistermaat, Grünbaum and Wilson.

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  1. Centre de recherches mathématiques, Université de Montréal, CP 6128, succ. Centre-Ville, Montréal, Quebe, H3C 3J7, Canada

    Yuri Berest

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Berest, Y. Hierarchies of Huygens' Operators and Hadamard's Conjecture. Acta Applicandae Mathematicae 53, 125–185 (1998). https://doi.org/10.1023/A:1006069012474

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