Structure-Preserving Flows of Symplectic Matrix Pairs

TL;DR

This paper introduces a nonlinear differential equation flow for symplectic matrix pairs that preserves structure and eigenvectors, governed by a Riccati differential equation, with explicit solutions and domain extension via Grassmann manifolds.

Contribution

It constructs a novel structure-preserving flow for symplectic matrix pairs, providing explicit solutions and methods to extend the flow beyond blow-up points.

Findings

Flow preserves symplectic structure and invariant subspaces.

Explicit solutions derived using Radon's lemma.

Domain extension achieved through Grassmann manifolds.

Abstract

We construct a nonlinear differential equation of matrix pairs $(\mathcal{M}(t),\mathcal{L}(t))$ that is invariant (the \textbf{Structure-Preserving Property}) in the class of symplectic matrix pairs \begin{align*} \mathbb{S}_{\mathcal{S}_1,\mathcal{S}_2}=\left\{\left(\mathcal{M},\mathcal{L}\right)| \ \mathcal{M}=\left[% \begin{array}{cc} X_{12} & 0 X_{22} & I \end{array}% \right]\mathcal{S}_2, \mathcal{L}=\left[% \begin{array}{cc} I & X_{11} 0 & X_{21} \end{array}% \right]\mathcal{S}_1\right.\nonumber \left. \text{ and }X=\left[% \begin{array}{cc} X_{11} & X_{12} X_{21} & X_{22} \end{array}% \right]\text{is Hermitian}\right\} \end{align*} for certain fixed symplectic matrices $\mathcal{S}_1$ and $\mathcal{S}_2$. Its solution also preserves invariant subspaces on the whole orbit (the \textbf{Eigenvector-Preserving Property}). Such a flow is called a \textit{structure-preserving flow} and is governed by a Riccati differential equation (RDE). In addition, Radon's lemma leads to an explicit form. Therefore, blow-ups for the structure-preserving flows may happen at a finite $t$. To continue, we then utilize the Grassmann manifolds to extend the domain of the structure-preserving flow to the whole $\mathbb{R}$ subtracting some isolated points.