Abstract
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 {Hk}. 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.
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