- A_D__modules (3)
Submitted manuscript, 417 KB, PDF document

Research output: Working paper

Published

2022.

Research output: Working paper

Blower, G & Doust, I 2022 'Linear systems, Hankel products and the sinh-Gordon equation'.

Blower, G., & Doust, I. (2022). *Linear systems, Hankel products and the sinh-Gordon equation*.

Blower G, Doust I. Linear systems, Hankel products and the sinh-Gordon equation. 2022 Oct 13.

@techreport{22fe6a19e2d24c0cb6b7e49ed0edbff3,

title = "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 $\phi_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator $\Gamma_{\phi_{(x)}}$; if $\Gamma_{\phi_{(x)}}$ is trace class, then the Fredholm determinant $\tau (x)=\det (I+\Gamma_{\phi_{(x)}})$ 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 Painl\'eve $\mathrm{III}'$ 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. ",

keywords = "Sinh-Gordon equation, Howland operators, tau functions, linear systems",

author = "Gordon Blower and Ian Doust",

year = "2022",

month = oct,

day = "13",

language = "English",

type = "WorkingPaper",

}

TY - UNPB

T1 - Linear systems, Hankel products and the sinh-Gordon equation

AU - Blower, Gordon

AU - Doust, Ian

PY - 2022/10/13

Y1 - 2022/10/13

N2 - 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 $\phi_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator $\Gamma_{\phi_{(x)}}$; if $\Gamma_{\phi_{(x)}}$ is trace class, then the Fredholm determinant $\tau (x)=\det (I+\Gamma_{\phi_{(x)}})$ 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 Painl\'eve $\mathrm{III}'$ 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.

AB - 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 $\phi_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator $\Gamma_{\phi_{(x)}}$; if $\Gamma_{\phi_{(x)}}$ is trace class, then the Fredholm determinant $\tau (x)=\det (I+\Gamma_{\phi_{(x)}})$ 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 Painl\'eve $\mathrm{III}'$ 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.

KW - Sinh-Gordon equation

KW - Howland operators

KW - tau functions

KW - linear systems

M3 - Working paper

BT - Linear systems, Hankel products and the sinh-Gordon equation

ER -