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Projective differential geometry of higher reductions of the two-dimensional Dirac equation | | L. V. Bogdanov
; E. V. Ferapontov
; | | Date: |
22 Nov 2002 | | Subject: | Exactly Solvable and Integrable Systems; Differential Geometry | nlin.SI math.DG | | Abstract: | 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 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 $P^{2n-1}$, 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. 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. | | Source: | arXiv, nlin.SI/0211040 | | Services: | Forum | Review | PDF | Favorites | |
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