Linear systems, Hankel products and the sinh-Gordon equation

Blower, Gordon and Doust, Ian (2022) Linear systems, Hankel products and the sinh-Gordon equation. Working Paper. UNSPECIFIED.

[thumbnail of A_D__modules (3)]
Text (A_D__modules (3))
A_D_modules_3_.pdf - Submitted Version

Download (426kB)


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.

Item Type:
Monograph (Working Paper)
?? sinh-gordon equationhowland operatorstau functionslinear systems ??
ID Code:
Deposited By:
Deposited On:
15 Nov 2022 15:15
Last Modified:
15 Jul 2024 08:01