论文标题
线性系统，Hankel产品和SINH-GORDON方程
Linear systems, Hankel products and the sinh-Gordon equation
论文作者
论文摘要
令$（-a，b，c）$是连续时间的线性系统$ t> 0 $，带有输入和输出空间$ {\ Mathbb C}^2 $和状态空间$ h $。散射功能$ ϕ _ {（x）}（t）= ce^{ - （t+2x）a} b $确定hankel积分运算符$γ_{ϕ _ {（x）}} $;如果$γ_{ϕ _ {（x）}} $是跟踪类，则fredholm necarent $τ（x）= \ det（i+γ_{ϕ _ {（x）}}）$确定$（ - a，b，c）$的tau函数。该论文建立了代数的属性，包括$ r_x = \ int_x^\ infty e^{ - ta} bce^{ - ta} dt $ on $ h $。因此，本文获得了Sinh-Gordon PDE的解决方案。 SINH-GORDON的tau功能满足特定的painléve$ \ mathrm {iii}'$ nonlinear ode，并描述了一个随机矩阵模型，库仑流体方法在间隔上是静电变异问题的解决方案。
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 $ϕ_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator $Γ_{ϕ_{(x)}}$; if $Γ_{ϕ_{(x)}}$ is trace class, then the Fredholm determinant $τ(x)=\det (I+Γ_{ϕ_{(x)}})$ 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 Painléve $\mathrm{III}'$ 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.
