  Based on an extended Skyrme interaction that includes the terms in relative
momenta up to sixth order, corresponding to the so-called Skyrme
pseudopotential up to next-to-next-to-next-to leading order (N3LO), we derive
the expressions of Hamiltonian density and single nucleon potential under
general non-equilibrium conditions which can be applied in transport model
simulations of heavy-ion collisions induced by neutron-rich nuclei. While the
conventional Skyrme interactions, which include the terms in relative momenta
up to second order, predict an incorrect behavior as a function of energy for
nucleon optical potential in nuclear matter, the present extended N3LO Skyrme
interaction can give a nice description for the empirical nucleon optical
potential. We also construct three interaction sets with different high-density
behaviors of the symmetry energy, by fitting both the empirical nucleon optical
potential up to energy of $1$ GeV and the empirical properties of isospin
asymmetric nuclear matter. These extended N3LO Skyrme interactions will be
useful in transport model simulations of heavy-ion collisions induced by
neutron-rich nuclei at intermediate and high energies, and they can also be
useful in nuclear structure studies within the mean-field model.
