Integrability of differential-difference equations with discrete kinks

We discuss a series of models introduced by Barashenkov, Oxtoby, and Pelinovsky to describe some discrete approximations of the Φ⁴ theory that preserve traveling kink solutions. Using the multiple scale test, we show that they have some integrability properties because they pass the A₁ and A₂ conditions, but they are nonintegrable because they fail the A₃ conditions.