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We study localized modes (LMs) of the one-dimensionalGross-Pitaevskii/nonlinear Schr\"{o}dinger equation with a harmonic-oscillator(parabolic) confining potential, and a periodically modulated coefficient infront of the cubic term (nonlinear lattice pseudopotential). The equationapplies to a cigar-shaped Bose-Einstein condensate loaded in the combination ofa magnetic trap and an optical lattice which induces the periodicpseudopotential via the Feshbach resonance. Families of stable LMs in the modelfeature specific properties which result from the interplay between spatialscales introduced by the parabolic trap and the period of the nonlinearpseudopotential. Asymptotic results on the shapes and stability of LMs areobtained for small-amplitude solutions and in the limit of a rapidlyoscillating nonlinear pseudopotential. We show that the presence of the latticepseudopotential may result in: (i) creation of new LM families which have nocounterparts in the case of the uniform nonlinearity; (ii) stabilization ofsome previously unstable LM species; (iii) evolution of unstable LMs into apulsating mode trapped in one well of the lattice pseudopotential.

Localized modes in the Gross-Pitaevskii equation with a parabolic trapping potential and a nonlinear lattice pseudopotential