Quaternionic Soliton Equations from Hamiltonian Curve Flows in HP^n

TL;DR

This paper derives a hierarchy of quaternionic soliton equations from geometric flows in quaternionic projective space, revealing new integrable structures and geometric interpretations of these equations.

Contribution

It introduces a novel bi-Hamiltonian hierarchy of quaternionic soliton equations from geometric curve flows in HP^n, including quaternionic sine-Gordon and mKdV equations.

Findings

Derived quaternionic sine-Gordon and mKdV equations with bi-Hamiltonian structure

Established geometric interpretations as non-stretching wave and Schrödinger maps

Connected quaternionic soliton equations to symmetric space geometry

Abstract

A bi-Hamiltonian hierarchy of quaternion soliton equations is derived from geometric non-stretching flows of curves in the quaternionic projective space $HP^n$. The derivation adapts the method and results in recent work by one of us on the Hamiltonian structure of non-stretching curve flows in Riemannian symmetric spaces $M=G/H$ by viewing $HP^n \simeq {\rm U}(n+1,H)/{\rm U}(1,H) \times {\rm U}(n,H)\simeq {\rm Sp}(n+1)/{\rm Sp}(1)\times {\rm Sp}(n)$ as a symmetric space in terms of compact real symplectic groups and quaternion unitary groups. As main results, scalar-vector (multi-component) versions of the sine-Gordon (SG) equation and the modified Korteveg-de Vries (mKdV) equation are obtained along with their bi-Hamiltonian integrability structure consisting of a shared hierarchy of quaternionic symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in $HP^n$ are shown to be described by a non-stretching wave map and a mKdV analog of a non-stretching Schrodinger map.