From the special 2 + 1 Toda lattice to the Kadomtsev - Petviashvili equation

The nonlinearization of the eigenvalue problems associated with the Toda hierarchy and the coupled Korteweg - de Vries (KdV) hierarchy leads to an integrable symplectic map S and an integrable Hamiltonian system (H₀), respectively. It is proved that S and (H₀) have the same integrals [H_k]. The quasi-periodic solution of the (2 + 1)-dimensional Kadomtsev - Petviashvili equation is split into three Hamiltonian systems (H₀), (H₁), (H₂), while that of the special (2 + 1)-dimensional Toda equation is separated into (H₀), (H₁) plus the discrete flow generated by the symplectic map S. A clear evolution picture of various flows is given through the 'window' of Abel - Jacobi coordinates. The explicit theta-function solutions are obtained by resorting to this separation technique.