  The Ablowitz-Ladik system, being one of the few integrable nonlinear
lattices, admits a wide class of analytical solutions, ranging from exact
spatially localised solitons to rational solutions in the form of the
spatiotemporally localised discrete Peregrine soliton. Proving a closeness
result between the solutions of the Ablowitz-Ladik and a wide class of Discrete
Nonlinear Schr\"odinger systems in a sense of a continuous dependence on their
initial data, we establish that such small amplitude waveforms may be supported
in the nonintegrable lattices, for significant large times. The nonintegrable
systems exhibiting such behavior include a generalisation of the Ablowitz-Ladik
system with a power-law nonlinearity and the Discrete Nonlinear Schr\"odinger
with power-law and saturable nonlinearities. The outcome of numerical
simulations illustrates in an excellent agreement with the analytical results
the persistence of small amplitude Ablowitz-Ladik analytical solutions in all
the nonintegrable systems considered in this work, with the most striking
example being that of the Peregine soliton.
