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 (H-0), respectively. It is proved that S and (H-0) have the same integrals {H-k} The quasi-periodic solution of the (2 + 1)-dimensional Kadomtsev-Petviashvili equation is split into three Hamiltonian systems (H-0), (H-1), (H-2), while that of the special (2 + 1)-dimensional Toda equation is separated into (H-0), (H-1) plus the discrete Bow 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.