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

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 $φ{(x)}(t)=Ce^{-(t+2x)A}B$ 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 $H$, 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 $KP$.