Integrable Systems with Unitary Invariance from Non-stretching Geometric Curve Flows in the Hermitian Symmetric Space Sp(n)/U(n)

TL;DR

This paper derives new integrable systems with unitary invariance from non-stretching geometric curve flows in the Hermitian symmetric space Sp(n)/U(n), utilizing a moving frame method and complex matrix representations.

Contribution

It introduces a novel application of the moving frame method to derive integrable systems in Sp(n)/U(n), including a bi-Hamiltonian mKdV and a Hamiltonian sine-Gordon equation.

Findings

Derived a bi-Hamiltonian modified KdV equation with unitary invariance.

Formulated a Hamiltonian sine-Gordon equation coupled with a scalar and complex vector.

Utilized the Hermitian structure of Sp(n)/U(n) for natural complex matrix representation.

Abstract

A moving parallel frame method is applied to geometric non-stretching curve flows in the Hermitian symmetric space Sp(n)/U(n) to derive new integrable systems with unitary invariance. These systems consist of a bi-Hamiltonian modified Korteweg-de Vries equation and a Hamiltonian sine-Gordon (SG) equation, involving a scalar variable coupled to a complex vector variable. The Hermitian structure of the symmetric space Sp(n)/U(n) is used in a natural way from the beginning in formulating a complex matrix representation of the tangent space sp(n)/u(n) and its bracket relations within the symmetric Lie algebra (u(n),sp(n)).