Huygens' Principle for Dirac Operators
Fabio Augusto da Costa Carvalho Chalub
Huygens' Principle | Dirac Operators | Integrability
This work is concerned with Huygens' property in Hadarmard's strict sense for Dirac operators. By such property we mean that the solutions of the initial value problem depend only on the intersection of the light cone with the initial data manifold, and not on the interior part of the light cone. A fascinating connection between the rational solutions of the Korteweg-de Vries hierarchy and Huygens property was discovered as a consequence of the independent works of Lagnese & Stellmacher and Adler & Moser. This link was further strenghened by the results of Y. Berest in 90's. The KdV equation was the tip of the iceberg for the rich theory of infinite dimensional completely integrable systems, which was later extended tremendously by Ablowitz, Kaup, Newell & Segur (AKNS) and Zakharov \& Shabat. In this work we establish a connection between the rational solutions of the AKNS hierarchy of nonlinear integrable equations and Dirac operators satisfying Huygens' property. We also characterize Huygens' operators in 1+1 dimensions and 3+1 under certain assumptions.