  The modified discrete nonlinear Schr\"odinger equation is used to study the
formation of stationary localized states in a one-dimensional lattice with a
single impurity and an asymmetric dimer impurity. A periodically modulated and
a perfectly nonlinear chain is also considered. Phase diagrams of localized
states for all systems are presented. From the mean square displacement
calculation, it is found that all states are not localized even though the
system comprises random nonlinear site energies. Stability of the states is
discussed.
