  We propose a Hamiltonian of ultracold spinless atoms in optical lattices
including the two-body interaction of nearest neighbors, which reduces to the
Bose-Hubbard model in weak interaction limit. An atom-pair hoping term
appearing in the new Hamiltonian explains naturally the recent experimental
observation of correlated tunneling in a double-well trap with strong atom-atom
interactions and moreover leads to a new dynamic process of atom-pair tunneling
where strongly interacting atoms can tunnel back and forth as a fragmented
pair. Finally a new dynamics of oscillations induced by the atom-pair tunneling
is found in the strong interaction regime, where the Bose-Hubbard model gives
rise to the insulator state with fixed time-averaged value of
atom-occupation-number only. Quantum phase transitions between two quantum
phases characterized by degenerate and non-degenerate ground states are shown
to be coinciding with the Landau second-order phase transition theory. In the
system of finite atom-number the degeneracy of ground states can be removed by
quantum tunneling for the even-number of atoms but not for the odd-number.
