We have studied the dynamical behavior of large amplitude displacements on a discrete monatomic chain whose first and second nearest neighbor particles interact following an anharmonic pair polynomial potential of cubic and/or quartic type. Kink and breather nontopological soliton solutions are found to exist in the continuum limit and envelope solitons or dark solutions are obtained without any continuum restriction for their carrier waves. Numerical simulations are extensively presented showing the solitonlike behavior of these solutions during their propagation and interactions on the discrete chain. New, interesting nonlinear phenomena are observed due to competition between first and second nearest neighbor's coupling (splitting or blow-up of solitons). Discreteness effects are discussed.