Integrable system on partial isometries: a finite dimensional picture
Tomasz Goliński, Alice Barbora Tumpach
TL;DR
This work develops a finite-dimensional realization of an integrable bihamiltonian hierarchy associated with the restricted Grassmannian by restricting to $N\\times M$ matrices and focusing on partial isometries from $\\mathcal{H}_+$ to $\\mathcal{H}_-$. The dynamics are given by Lax flows $\\frac{\\partial}{\\partial t_k^n}\\mu = i^{n+1}[\\mu, H_k^n(\\mu)]$, and under $\\mu_{++}=0$ the evolution reduces to a flow on partial isometries $u$, with $\\partial_{t^n_k}u = i^{n+1}(\\mu H^{n-1}_{k-1}(\\mu))_{--}u$; explicit rank-one solutions exhibit decoupled phase dynamics. In the 2+2 model one obtains fully explicit formulas for the amplitudes and phases, illustrating solvability of the finite-dimensional picture and suggesting extensions to higher rank and optical-system contexts.
Abstract
The aim of the paper is to present the integrable systems on partial isometries which are related to the restricted Grassmannian in finite dimensional context. Some explicit solutions are obtained.