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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)
##### Keywords:
para-Kähler structure; noncommutative KdV equation
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