  For slowly-varying initial data, solutions to the Ablowitz-Ladik system have
been proven to converge to solutions of the cubic Schr\"odinger equation. In
this paper we show that in the continuum limit, solutions to the Ablowitz-Ladik
system with $H^1$ initial data may also converge to solutions of the modified
Korteweg--de Vries equation. To exhibit this new limiting behavior, it suffices
that the initial data is supported near the inflection points of the dispersion
relation associated with the Ablowitz-Ladik system.
  Our arguments employ harmonic analysis tools, Strichartz estimates, and the
conservation of mass and energy. Correspondingly, they are applicable beyond
the completely integrable models of greatest interest to us.
