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Projective differential geometry of higher reductions of the two-dimensional Dirac equation by L.V. Bogdanov; E.V. Ferapontov is a Mathematics article available to read on EtoBox.
What is Projective differential geometry of higher reductions of the two-dimensional Dirac equation about?
We investigate reductions of the two-dimensional Dirac equation imposed by the requirement of the existence of a differential operator D n of order n mapping its eigenfunctions to adjoint eigenfunctions. For first order operators these reductions (and multicomponent analogs thereof) lead to the Lame equations descriptive of orthogonal coordinate systems. Our main observation is that nth order reductions coincide with the projective-geometric 'Gauss-Codazzi' equations governing special classes of line congruences in the projective space P 2n-1 , which is the projectivised kernel of D n . In the second order case this leads to the theory of W-congruences in P 3 which belong to a linear complex, while the third order case corresponds to isotropic congruences in P 5 . Higher reductions are compatible with odd order flows of the Davey-Stewartson hierarchy. All these flows preserve the kernel D n , thus defining nontrivial geometric evolutions of line congruences. Multicomponent 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 o
Who reads Projective differential geometry of higher reductions of the two-dimensional Dirac equation?
It is typically read by researchers, students, and practitioners in Mathematics.
- Author
- L.V. Bogdanov; E.V. Ferapontov
- Publisher
- Elsevier Science; Elsevier ; Elsevier BV (ISSN 0393-0440)
- Published
- 2004
- Language
- EN
- Field
- Mathematics (Physical Sciences)