Theoretical and numerical analysis of a degenerate nonlinear cubic Schrödinger equation

Abstract

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).

Keywords

• cubic schrödinger equation
• time-dependent schrödinger equation
• interior degeneracy
• cranknicolson method
• galerkin method
• 35k65
• 35q40
• 35q41
• 35j10
• 65p40