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 C2 and state space H. The scattering (or impulse response) functions ϕ(x)(t)=Ce−(t+2x)AB determines a Hankel integral operator Γϕ(x); if Γϕ(x) is trace class, then the Fredholm determinant τ(x)=det⁡(I+Γϕ(x)) determines the tau function of (−A,B,C). The paper establishes properties of algebras containing Rx=∫x∞e−tABCe−tAdt on H, and obtains solutions of the sinh-Gordon PDE. The tau function for sinh-Gordon satisfies a particular Painlevé 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.