Theoretical and numerical analysis of a degenerate nonlinear cubic Schrödinger equation

Abstract

<jats:title>Abstract</jats:title>
<jats:p>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 ∃<jats:italic>x</jats:italic>
<jats:sub>0</jats:sub> ∈ (0, <jats:italic>L</jats:italic>), <jats:italic>s</jats:italic>. <jats:italic>t k</jats:italic>(<jats:italic>x</jats:italic>
<jats:sub>0</jats:sub>) = 0, where <jats:italic>k</jats:italic> stands for the diffusion coefficient and <jats:italic>L</jats:italic> 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 <jats:italic>k</jats:italic> namely <jats:italic>k</jats:italic>(<jats:italic>x</jats:italic>) = |<jats:italic>x</jats:italic> − <jats:italic>x</jats:italic>
<jats:sub>0</jats:sub>| <jats:sup>α</jats:sup>, <jats:italic>α</jats:italic> ∈ (0, 1).</jats:p>