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We address the Gross--Pitaevskii (GP) equation with a periodic linearpotential and a periodic sign-varying nonlinearity coefficient. Contrary to theclaims in the previous works of Abdullaev {\em et al.} [PRE {\bf 77}, 016604(2008)] and Smerzi & Trombettoni [PRA {\bf 68}, 023613 (2003)], we show thatthe intersite cubic nonlinear terms in the discrete nonlinear Schr\"odinger(DNLS) equation appear beyond the applicability of assumptions of thetight-binding approximation. Instead of these terms, for an even linearpotential and an odd nonlinearity coefficient, the DNLS equation and otherreduced equations for the semi-infinite gap have the quintic nonlinear term,which correctly describes bifurcation of gap solitons.

Bifurcation of gap solitons in periodic potentials with a sign-varying nonlinearity coefficient