Linear systems, Hankel products and the sinh-Gordon equation

Blower, Gordon and Doust, Ian (2023) Linear systems, Hankel products and the sinh-Gordon equation. Journal of Mathematical Analysis and Applications, 525 (1). ISSN 0022-247X

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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 (or impulse response) 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 containing $R_x = \int_x^\infty e^{-tA}BCe^{-tA}\,dt$ on $H$, and 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.