Linear systems, Hankel products and the sinh-Gordon equation
Linear systems, Hankel products and the sinh-Gordon equation
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DOI：
10.1016/j.jmaa.2023.127140
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发表时间：
2022-10
影响因子：
1.3
通讯作者：
G. Blower;I. Doust
中科院分区：
文献类型：
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作者：
G. Blower;I. Doust
设（− A，B，C）是连续时间t> 0的线性系统，其输入输出空间为C2，状态空间为H.散射（或脉冲响应）函数B（x）（t）= C e−（t+ 2 x）A确定一个汉克尔积分算子Γ（x）;如果Γ（x）是迹类，则Fredholm行列式τ（x）= det（I+ Γ（x））确定（− A，B，C）的τ函数。建立了H上包含Rx = Rx ∞ e− B − tAdt的代数的性质，并得到了sinh-Gordon偏微分方程的解. sinh-Gordon的τ函数满足一个特殊的Painlevé III′非线性常微分方程，描述了一个随机矩阵模型，其渐近分布由库仑流体方法发现是一个区间上的静电变分问题的解.
Abstract Let (− A, B, C) be a linear system in continuous time t> 0 with input and output space C 2 and state space H. The scattering (or impulse response) functions ϕ (x)(t)= C e−(t+ 2 x) 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 containing R x=∫ x∞ e− t A B C e− t A d t 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.