Unitarily-invariant integrable systems and geometric curve flows in $SU(n+1)/U(n)$ and $SO(2n)/U(n)$

TL;DR

This paper derives new integrable multi-component soliton equations from geometric curve flows in Hermitian symmetric spaces, revealing their bi-Hamiltonian structures and geometric interpretations.

Contribution

It introduces novel multi-component integrable equations and their geometric origins in symmetric spaces, expanding the understanding of soliton hierarchies.

Findings

New multi-component Sine-Gordon and mKdV equations

A nonlocal multi-component nonlinear Schrödinger equation

Bi-Hamiltonian structures and recursion operators for these systems

Abstract

Bi-Hamiltonian hierarchies of soliton equations are derived from geometric non-stretching (inelastic) curve flows in the Hermitian symmetric spaces $SU(n+1)/U(n)$ and $SO(2n)/U(n)$. The derivation uses Hasimoto variables defined by a moving parallel frame along the curves. As main results, new integrable multi-component versions of the Sine-Gordon (SG) equation and the modified Korteveg-de Vries (mKdV) equation, as well as a novel nonlocal multi-component version of the nonlinear Schr\"odinger (NLS) equation are obtained, along with their bi-Hamiltonian structures and recursion operators. These integrable systems are unitarily invariant and correspond to geometric curve flows given by a non-stretching wave map and a mKdV analog of a non-stretching Schr\"odinger map in the case of the SG and mKdV systems, and a generalization of the vortex filament bi-normal equation in the case of the NLS systems.