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Hierarchies of Huygens' Operators and Hadamard's Conjecture

机译:惠更斯算子的层次和哈达玛猜想

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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-Backlund transformations and the bispectral problem in the sense of Duistermaat, Grunbaum and Wilson.
机译:对于J. Hadamard意义上满足Huygens原理的线性双曲偏微分算子,我们开发了一种新的统一方法。基本的方法本质上是代数的,并且基于解析微分算子环中相似性(规范)转换的某种非线性扩展。本文提供了关于Minkowski空间中惠更斯原理的经典和最新结果的系统且自洽的综述。这些结果大多数以平面重力波的形式延续到更一般的伪黎曼空间。惠更斯原理与可积系统和孤子理论的各种联系都得到了特别的关注。在Duistermaat,Grunbaum和Wilson的意义上,我们讨论了与非线性KdV型演化方程,Darboux-Backlund变换和双谱问题的联系。

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