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Grassmannians Gr(N-1,N+1), closed differential N-1 forms and N-dimensional integrable systems | | L. V. Bogdanov
; B. G. Konopelchenko
; | | Date: |
30 Aug 2012 | | Abstract: | Integrable flows on the Grassmannians Gr(N-1,N+1) are defined by the
requirement of closedness of the differential N-1 forms $Omega_{N-1}$ of rank
N-1 naturally associated with Gr(N-1,N+1). Gauge-invariant parts of these
flows, given by the systems of the N-1 quasi-linear differential equations,
describe coisotropic deformations of (N-1)-dimensional linear subspaces. For
the class of solutions which are Laurent polynomials in one variable these
systems coincide with N-dimensional integrable systems such as Liouville
equation (N=2), dispersionless Kadomtsev-Petviashvili equation (N=3),
dispersionless Toda equation (N=3), Plebanski second heavenly equation (N=4)
and others. Gauge invariant part of the forms $Omega_{N-1}$ provides us with
the compact form of the corresponding hierarchies. Dual quasi-linear systems
associated with the projectively dual Grassmannians Gr(2,N+1) are defined via
the requirement of the closedness of the dual forms $Omega_{N-1}^{star}$. It
is shown that at N=3 the self-dual quasi-linear system, which is associated
with the harmonic (closed and co-closed) form $Omega_{2}$, coincides with the
Maxwell equations for orthogonal electric and magnetic fields. | | Source: | arXiv, 1208.6129 | | Services: | Forum | Review | PDF | Favorites | |
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