Linear systems, determinants and solutions of the Kadomtsev-Petviashvili equation
Gordon Blower, Simon J. Malham
TL;DR
This work connects Kadomtsev–Petviashvili equation solutions with Fredholm determinants of Hankel operators built from linear-system impulse responses. It develops a Lyapunov-based $R_x$ framework in which $\tau(x)=\det(I+R_x)$ yields KP potentials via Gelfand–Levitan–Marchenko-type relations, and embeds this in Pöppe's semi-additive operator theory with a Fedosov-product differential ring to enable efficient tau-function computation. It also provides a robust operator-theoretic foundation through spectral theory for cosine families, allowing a non-self-adjoint extension of the Gelfand–Levitan approach. Numerically, the approach underpins determinant-based and GLM-based schemes (Clenshaw–Curtis, Nyström methods) for computing KP solutions, including two-soliton scenarios. The results advance integrable-systems practice by unifying tau-function formalism, differential rings, and operator-spectral methods for explicit KP solutions.
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 $KP$.