Matrix Bispectrality and Huygens' Principle for Dirac Operators
Jorge Zubelli | Chalub, Fabio
Huygens principle; Dirac Operators; Bispectrality; AKNS
We explore relations among Huygens' principle for Dirac operators, rational solutions of the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, and matrix bispectrality. We show how such properties are connected by relating the Hadamard expansion coefficients to the expressions of the nonlinear AKNS flows. The matrix properties above have natural scalar counterparts obtained by reduction to certain manifolds, in which case we get Huygens' principle for wave operators, rational solutions of the Korteweg-de Vries equation and scalar bispectrality. As a by-product we give an alternative proof to a classical result of Lagnese and Stellmacher.