Projective differential geometry of higher reductions of the two-dimensional Dirac equation

الأصل · EN

We investigate reductions of the two-dimensional Dirac equation imposed by the requirement of the existence of a differential operator Dₙ of order n mapping its eigenfunctions to adjoint eigenfunctions. For first order operators these reductions (and multi-component analogs thereof) lead to the Lame equations descriptive of orthogonal coordinate systems. Our main observation is that n-th order reductions coincide with the projective-geometric `Gauss-Codazzi' equations governing special classes of line congruences in the projective space P²ⁿ⁻¹, which is the projectivised kernel of Dₙ. In the second order case this leads to the theory of W-congruences in P³ which belong to a linear complex, while the third order case corresponds to isotropic congruences in P⁵. Higher reductions are compatible with odd-order flows of the Davey-Stewartson hierarchy. All these flows preserve the kernel Dₙ, thus defining nontrivial geometric evolutions of line congruences. Multi-component generalizations are also discussed. The correspondence between geometric picture and the theory of integrable systems is established; the definition of the class of reductions and all geometric objects in terms of the multicomponent KP hierarchy is presented. Generating forms for reductions of arbitrary order are constructed.