An elementary approach to the polynomial \(\tau\)-functions of the KP-hierarchy (Q1586695)

Solutions of the KP hierarchy associated with polynomial \(\tau\)-functions are recovered in an elementary way, using a geometric approach to the linearization of the KP equations. It starts from the geometry of bi-Hamiltonian manifolds and allows obtaining a system of infinite-matrix Riccati equations, which are linearized by elementary methods. Here the authors explicitly determine the solutions of the Sato system where the initial condition has only a finite number of nonzero entries, and show that these solutions yield solutions of the KP hierarchy associated with polynomial \(\tau\)-functions.