  This is the second part of a series of papers where we develop rigorous decay
estimates for breather solutions of an averaged version of the non-linear
Schr\"odinger equation. In this part we study the diffraction managed discrete
non-linear Schr\"odinger equation, an equation which describes coupled
waveguide arrays with periodic diffraction management geometries. We show that,
for vanishing average diffraction, all solutions of the non-linear and
non-local diffraction management equation decay super-exponentially. As a
byproduct of our method, we also have a simple proof of existence of
diffraction managed solitons in the case of vanishing average diffraction.
