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 ${\varphi }_{(x)}(t)=C{e}^{-(t+2x)A}B$ determines a Hankel integral operator ${\Gamma }_{{\varphi }_{(x)}}$; if ${\Gamma }_{{\varphi }_{(x)}}$ is trace class, then the Fredholm determinant $\tau (x)=\det (I+{\Gamma }_{{\varphi }_{(x)}})$ 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.

References in corpus (5)

• Fredholm Determinants, Differential Equations and Matrix Models
• Entanglement and alpha entropies for a massive Dirac field in two dimensions
• Hermitian vs. Anti-Hermitian 1-Matrix Models and Their Hierarchies
• Fredholm determinants and the mKdV/sinh-Gordon hierarchies
• Grassmannian flows and applications to non-commutative non-local and local integrable systems

Cited by in corpus (1)

• Linear systems, spectral curves and determinants