Solitons in one-dimensional (1D) Bose-Einstein condensates with attractive interaction in variable anharmonic periodic potentials are presented. These variable periodic potentials allow us to consider a variety of potential shapes between the two special limits of the Kronig-Penney model and the inverse Kronig-Penney model, with the sinusoidal optical lattice as an intermediate case. The Gross-Pitaevskii equation in (1D) is taken as a model [53]. We have derived the stability conditions for BECs in variable anharmonic periodic potentials using the Vakhitov-Kolokolov criterion. We have found that the stability condition for solitons depends significantly on the shape parameter. In particular, depending on the values of the shape parameter, one and three localized states for the 1D case, respectively, have been found. We have shown that some of these localized states are stable or unstable. The stability conditions obtained are an extension of the existing results to: (1) the more general class of external potential, and (2) the Vakhitov-Kolokolov criterion has been generalized in the case of an external potential of arbitrary shape. A fully numerical simulation of the 1D Gross-Pitaevskii equation finally tests the results of the variational approximation. A good agreement between both the methods is observed.

Bose-Einstein condensate, Gross-Pitaevskii equation, solitons, stability criterion, variable anharmonic periodic potentials.