Huygens' Principle for Hyperbolic Operators and Integrable Hierarchies
Jorge P. Zubelli | Chalub, Fabio A.C.C.
Huygens' Principle | Dirac Operators | Rational Solutions of Integrable Equations.
We show that the stationary solutions of the canonical AKNS hierarchy of nonlinear evolution equations yield perturbations of Dirac operators that satisfy a strict form of Huygens' principle. Namely, the domain of dependence of such Dirac operators at a given point $y$ is contained in the light-cone's hypersurface issued from $y$. By canonical AKNS hierarchy we mean that the differential polynomials defining the flows are isobaric with respect to certain weights. The method we employ is of interest by itself. Indeed, we consider the Riesz kernels associated to a given hyperbolic differential operator and expand the fundamental solution of perturbations of this operator in a series in such Riesz kernels. Using the coefficients of this Hadamard type expansion we introduce a family of vector fields. For the D'Alembertian such vector field family corresponds to the KdV hierarchy and for the Dirac operators they include the AKNS one.