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We investigate adiabatic evolution of quantum states as governed by the nonlinear Schro¨dinger equation and provide examples of applications with a nonlinear tunneling model for Bose-Einstein condensates.Our analysis not only spells out conditions for adiabatic evolution of eigenstates but also characterizes the motion of noneigenstates which cannot be obtained from the former in the absence of the superposition principle.We also investigate the Berry phase acquired by an eigenstate that experienced a nonlinear adiabatic evolution.The circuit integral of the Berry connection of the instantaneous eigenstate cannot account for the adiabatic geometric phase,while the Bogoliubov excitations around the eigenstates are found to be accumulated during the nonlinear adiabatic evolution and contribute a finite phase of geometric nature.Some possible applications of our theory are discussed.