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

Abstract

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.