On the existence of dark solitons of the defocusing cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity

We provide a simple proof of the existence of dark solitons of the defocusing cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity. Moreover, our proof allows for a broader class of inhomogeneities and gives some new properties of the solutions. We also apply our approach to the defocusing cubic-quintic nonlinear Schrödinger equation with a periodic potential. We consider the following defocusing cubic nonlinear Schrödinger equation with inhomogeneous nonlinearity on R: iψt = − 1 2 ψxx + g(x)|ψ|ψ, (0.1) with g reasonably smooth, g being T − periodic (0.2) and satisfying 0 < gmin ≤ g(x) ≤ gmax. (0.3) The above problem arises in a variety of physical situations such as Bose-Einstein condensates, nonlinear optics etc (we refer the interested reader to [3, 4, 5] and the references therein for more details). The solitary wave solutions of (0.1) are given by ψ(x, t) = eφ(x), (0.4) where φ is real valued and solves − 1 2 φxx + λφ+ g(x)φ 3 = 0. (0.5) A solitary wave solution ψ of (0.1) is called a dark soliton if the associated φ satisfies φ(x) φ±(x) → 1 as x→ ±∞, (0.6) where the functions φ± are sign definite, T -periodic solutions of (0.5). It is worth mentioning that, in the case where g is a constant, the dark soliton can be represented explicitly and its stability has been a topic of extensive investigations in recent years (see [8] and the references therein). If λ ≥ 0, it was shown in [4] that the only bounded solution of (0.5) is the trivial one. Therefore, we will assume that λ < 0. (0.7)