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The noncommutative KdV equation and its para-Kähler structure. (English) Zbl 1302.37045
Summary: We prove that the noncommutative (\(n \times n\))-matrix KdV equation is exactly a reduction of the geometric KdV flows from \(\mathbb{R}\) to the symmetric para-Grassmannian manifold \(\tilde G_{2n,n}=\mathrm{SL}(2n,\mathbb R)/\mathrm{SL}(n,\mathbb R)\times\mathrm{SL}(n,\mathbb R)\) and it can also be realized geometrically as a motion of Sym-Pohlmeyer curves in the symmetric Lie algebra \(sl(2n,\mathbb R)\) with initial data being suitably restricted. This gives a para-geometric characterization of the noncommutative matrix KdV equation.

MSC:
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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