Editing Huygens–Fresnel principle (section)

==In other spatial dimensions==
In 1900, [[Jacques Hadamard]] observed that Huygens' principle was broken when the number of spatial dimensions is even.<ref>{{cite web |first=Alexander P. |last=Veselov |url=http://www.lboro.ac.uk/microsites/maths/research/preprints/papers02/02-49.pdf |title=Huygens' Principle |archive-url=https://web.archive.org/web/20160221215126/http://www.lboro.ac.uk/microsites/maths/research/preprints/papers02/02-49.pdf |archive-date=2016-02-21 |date=2002 }}</ref><ref>{{cite web |url=https://web.stanford.edu/class/math220a/handouts/waveequation3.pdf |title=Wave Equation in Higher Dimensions |publisher=Stanford University |work=Math 220a class notes }}</ref><ref>{{cite journal |first1=M. |last1=Belger |first2=R. |last2=Schimming |first3=V. |last3=Wünsch |title=A Survey on Huygens' Principle |journal=Zeitschrift für Analysis und ihre Anwendungen |volume=16 |issue=1 |date=1997 |pages=9–36 |doi=10.4171/ZAA/747 |doi-access=free }}</ref> From this, he developed a set of conjectures that remain an active topic of research.<ref>{{cite journal |first=Leifur |last=Ásgeirsson |author-link=Leifur Ásgeirsson |title=Some hints on Huygens' principle and Hadamard's conjecture |journal=Communications on Pure and Applied Mathematics |volume=9 |issue=3 |pages=307–326 |date=1956 |doi=10.1002/cpa.3160090304 }}</ref><ref>{{cite journal |first=Paul |last=Günther |title=Huygens' principle and Hadamard's conjecture |journal=The Mathematical Intelligencer |date=1991 |volume=13 |issue=2 |pages=56–63 |doi=10.1007/BF03024088 |s2cid=120446795 }}</ref> In particular, it has been discovered that Huygens' principle holds on a large class of [[homogeneous space]]s derived from the [[Coxeter group]] (so, for example, the [[Weyl group]]s of simple [[Lie algebra]]s).<ref name="veselov">{{cite journal |first=Alexander P. |last=Veselov |title=Huygens' principle and integrable systems |journal=Physica D: Nonlinear Phenomena |volume=87 |issue=1–4 |year=1995 |pages=9–13 |doi=10.1016/0167-2789(95)00166-2 |bibcode=1995PhyD...87....9V }}</ref><ref>{{cite journal |first1=Yu. Yu. |last1=Berest |first2=A. P. |last2=Veselov |title=Hadamard's problem and Coxeter groups: New examples of Huygens' equations |journal=Functional Analysis and Its Applications |date=1994 |volume=28 |issue=1 |pages=3–12 |doi=10.1007/BF01079005 |s2cid=121842251 }}</ref>

The traditional statement of Huygens' principle for the [[D'Alembertian]] gives rise to the [[KdV hierarchy]]; analogously, the [[Dirac operator]] gives rise to the [[AKNS]] hierarchy.<ref>{{cite journal |first1=Fabio A. C. C. |last1=Chalub |first2=Jorge P. |last2=Zubelli |title=Huygens' Principle for Hyperbolic Operators and Integrable Hierarchies |journal=Physica D: Nonlinear Phenomena |volume=213 |issue=2 |date=2006 |pages=231–245 |doi=10.1016/j.physd.2005.11.008 |bibcode=2006PhyD..213..231C }}</ref><ref>{{cite journal |first1=Yuri Yu. |last1=Berest |first2=Igor M. |last2=Loutsenko |title=Huygens' Principle in Minkowski Spaces and Soliton Solutions of the Korteweg-de Vries Equation |journal=Communications in Mathematical Physics |date=1997 |volume=190 |issue=1 |pages=113–132 |arxiv=solv-int/9704012 |doi=10.1007/s002200050235 |bibcode=1997CMaPh.190..113B |s2cid=14271642 }}</ref>