We study the propagation of vertical dust grain oscillations in dusty plasma crystals. Applying the quasidiscrete approximation on the model of the dusty plasma crystal [see I. Kourakis and P. K. Shukla, Phys. Plasmas 12, 014502 (2005)], we derive a nonlinear Schrödinger equation and find that the system supports bright and dark soliton solutions. Also, it appears contrary to existing work that the lattice has a zero-dispersion frequency which induces the resonance of the nonlinear coefficient. The numerical integration of the equation of the model and the energy density confirm the backward-wave predicted by the dispersion relation and the fact that the phase and group velocities are antiparallel.