Preprint A327/2004
Huygens' Principle for Hyperbolic Operators and Integrable Hierarchies

Jorge P. Zubelli | Chalub, Fabio A.C.C.

**Keywords: **
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.