Linear systems, Hankel products and the sinh-Gordon equation

TL;DR

This paper links linear systems, Hankel operators, and tau functions to solutions of the sinh-Gordon PDE, revealing new algebraic properties and connections to random matrix models and Painlevé equations.

Contribution

It establishes algebraic properties of operators associated with linear systems and derives solutions to the sinh-Gordon PDE using tau functions and Hankel operators.

Findings

Tau function satisfies Painlevé III' equation.

Solutions describe a random matrix model.

Asymptotic distribution obtained via Coulomb fluid method.

Abstract

Let $(-A,B,C)$ be a linear system in continuous time $t>0$ with input and output space ${\mathbb C}^2$ and state space $H$. The scattering functions $\phi_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator $\Gamma_{\phi_{(x)}}$; if $\Gamma_{\phi_{(x)}}$ is trace class, then the Fredholm determinant $\tau (x)=\det (I+\Gamma_{\phi_{(x)}})$ determines the tau function of $(-A,B,C)$. The paper establishes properties of algebras including $R_x=\int_x^\infty e^{-tA}BCe^{-tA}dt$ on $H$. Thus the paper obtains solutions of the sinh-Gordon PDE. The tau function for sinh-Gordon satisfies a particular Painl\'eve $\mathrm{III}'$ nonlinear ODE and describes a random matrix model, with asymptotic distribution found by the Coulomb fluid method to be the solution of an electrostatic variational problem on an interval.