Theoretical and numerical analysis of a degenerate nonlinear cubic Schrödinger equation

In this paper, we are interested in some theoretical and numerical studies of a special case of a degenerate nonlinear Schrödinger equation namely the so-called Gross-Pitaevskii Equation(GPE). More precisely, we will treat in a first time the well-posedness of GPE model with a degeneracy occurring in the interior of the space variable domain, i.e ∃x₀ ∈ (0, L), s. t k(x₀) = 0, where k stands for the diffusion coefficient and L is a positive constant. Thereafter, we will focus ourselves on some numerical simulations showing the influence of a different parameters, especially the interior degeneracy, on the behavior of the wave solution corresponding to our model in a special case of the function k namely k(x) = |x − x₀| ^α, α ∈ (0, 1).