Solutions of ultrahyperbolic equations and their application in texture analysis

Solutions of ultrahyperbolic $2\times 2$ equations are examined. An inverse diffraction problem—the recovery of the orientation distribution function from x-ray or neutron measurements of pole figures (PFs)—is reduced to a system of integral equations on the rotation group $SO(3)$. By transforming $SO(3)$, the original problem is reduced to a well-known problem in integral geometry, namely, to the recovery of a function in three-dimensional space from known integrals along straight lines. Upon this transform, the PFs are solutions to an ultrahyperbolic equation and the solution is nonunique.