Linear systems, determinants and solutions of the Kadomtsev-Petviashvili equation

TL;DR

This paper links linear systems and operator theory to solutions of the Kadomtsev-Petviashvili equation, introducing algebraic structures and determinants for effective numerical computation.

Contribution

It establishes new algebraic properties of operators related to the KP equation and connects Fredholm determinants with solution methods.

Findings

Fredholm determinants encode KP solutions

Operator algebras are characterized for solution analysis

Numerical methods for KP solutions are improved

Abstract

Let $(-A,B,C)$ be a linear system in continuous time $t>0$ 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. P\"oppe's semi-additive operators are identified with orbits of a shift action on integral kernels, and P\"oppe'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 $KP$.