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Linear systems, determinants, and solutions of the Kadomtsev–Petviashvili equation
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 21 July 2026, pp. 1-55
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- Article
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Let
$(-A,B,C)$ be a linear system in continuous time 
$t \gt 0$ with input and output space 
${\mathbb C}$ and state space 
$H$. The scattering (or impulse response) 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 
$H$, and obtains solutions of the Kadomtsev–Petviashvili PDE. Pöppe’s semi-additive operators are identified with orbits of a shift action on integral kernels, and Pöppe’s bracket operation is expressed in terms of the Fedosov product. The paper shows that the Fredholm determinant 
$\det (I+R_x)$ gives an effective method for numerical computation of solutions of 
$KP$.
