The Poisson structure on is introduced to give the Hamiltonian system associated with a spectral problem which yields the nonlinear Schrödinger (NLS) hierarchy. The Hamiltonian system is proven to be Liouville integrable. Some -dimensional equations including NLS equation, Kadomtesev–Petviashvili I (KPI) equation, coupled KPI equation, and modified Kadomtesev–Petviashvili (mKP) equation, are decomposed into Hamilton flows via the NLS hierarchy. The algebraic curve, Abel–Jacobi coordinates, and Riemann–Jacobi inversion are used to obtain the algebrogeometric solutions of these equations.
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