Strong non-monotonic behavior of particle density of solitary waves of nonlinear Schrödinger equation in Bose-Einstein condensates

We study the focusing nonlinear Schrödinger equation (NLSE) which generalizes 1D Gross-Pitaevskii equation (GPE) with attractive atom-atom spatially (in)homogeneous interaction in Bose-Einstein condensates, where the potential is a non-monotone function, periodic or not. Following some recently published numerically simulations of the particle density of solutions of GPE with periodic potentials, one can conclude, it admits the non-monotonic behavior with respect to the spatial variable. Here, we present a mathematical approach to justify that, by giving a constructive method and finding some conditions on chemical and external potentials such that the particle density of solitary wave of NLSE has sign-changing first derivative as a kind of strong non-monotonic behavior of positive function. We apply it to the GPE with non-periodic as well as periodic potential having small enough amplitude and frequency.