Linear systems, Hankel products and the sinh-Gordon equation

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 ${\phi }_{(x)}(t)=C{e}^{-(t+2x)A}B$ determines a Hankel integral operator ${\mathrm{\Gamma }}_{{\phi }_{(x)}}$; if ${\mathrm{\Gamma }}_{{\phi }_{(x)}}$ is trace class, then the Fredholm determinant $\tau (x)=\det (I+{\mathrm{\Gamma }}_{{\phi }_{(x)}})$ determines the tau function of $(-A,B,C)$. The paper establishes properties of algebras including $R_x={\int }_x^{\mathrm{\infty }}{e}^{-tA}BC{e}^{-tA}dt$ on $H$. Thus the paper obtains solutions of the sinh-Gordon PDE. The tau function for sinh-Gordon satisfies a particular Painléve ${\mathrm{I}\mathrm{I}\mathrm{I}}^{\mathrm{\prime }}$ 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.