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arXiv · 1206.4040

Symplectically-invariant soliton equations from non-stretching geometric curve flows

Stephen C. Anco·Esmaeel Asadi
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Abstract

A moving frame formulation of geometric non-stretching flows of curves in the Riemannian symmetric spaces $Sp(n+1)/Sp(1)\times Sp(n)$ and $SU(2n)/Sp(n)$ is used to derive two bi-Hamiltonian hierarchies of symplectically-invariant soliton equations. As main results, multi-component versions of the sine-Gordon (SG) equation and the modified Korteweg-de Vries (mKdV) equation exhibiting $Sp(1)\times Sp(n-1)$ invariance are obtained along with their bi-Hamiltonian integrability structure consisting of a shared hierarchy of symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in $Sp(n+1)/Sp(1)\times Sp(n)$ and $SU(2n)/Sp(n)$ are shown to be described by a non-stretching wave map and a mKdV analog of a non-stretching Schr\"odinger map.

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Stephen C. Anco, Esmaeel Asadi. 2012-06-18. Symplectically-invariant soliton equations from non-stretching geometric curve flows. https://doi.org/10.1088/1751-8113/45/47/475207

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