Linear systems, Hankel products and the sinh-Gordon equation

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