Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2002-11-22
Nonlinear Sciences
Exactly Solvable and Integrable Systems
LaTeX, 25 pages
Scientific paper
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.
Bogdanov L. V.
Ferapontov E. V.
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