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 $\ddot{I}{•}_{(x)}(t)=C{e}^{-(t+2x)A}B$ determines a Hankel integral operator $\hat{I}{“}_{\ddot{I}{•}_{(x)}}$; if $\hat{I}{“}_{\ddot{I}{•}_{(x)}}$ is trace class, then the Fredholm determinant $\ddot{I}„(x)=\det (I+\hat{I}{“}_{\ddot{I}{•}_{(x)}})$ determines the tau function of $(-A,B,C)$. The paper establishes properties of algebras including $R_x={\int }_x^{\infty }{e}^{-tA}BC{e}^{-tA}dt$ 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 $KP$.

55 pages, 4 figures