Modulational instability of Bose-Einstein condensate in a complex polynomial in elliptic Jacobian functions potential

We consider a Gross-Pitaevskii (GP) equation with cubic-quintic nonlinearities, which governs the dynamics of Bose-Einstein condensates (BECs) matter waves with time-dependent complex potential in Jacobian elliptic functions. The complex term of the potential accounts for either the atomic feeding or the atomic loss of the condensate. Based on a special variable transformation, an integrable condition is obtained and used, firstly, to explicitly express the growth rate of a purely growing modulational instability and, secondly, to derive classes of exact solitonic and periodic solutions. Analytical solitonic solutions describe the propagation of both dark and bright solitary waves of the BECs. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013