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

TL;DR

This paper derives new multi-component symplectically-invariant soliton equations from geometric curve flows in symmetric spaces, revealing their bi-Hamiltonian structure and geometric interpretations as wave and Schrödinger map analogs.

Contribution

It introduces two bi-Hamiltonian hierarchies of soliton equations with symplectic invariance derived from geometric flows in specific symmetric spaces, a novel connection between geometry and integrable systems.

Findings

Multi-component sine-Gordon and mKdV equations with Sp(1)×Sp(n-1) invariance

Bi-Hamiltonian integrability structure with hereditary recursion operator

Geometric curve flows described by wave and Schrödinger map analogs

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.