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

Theoretical and numerical analysis of a degenerate nonlinear cubic Schrödinger equation

Department of Mathematics, Faculty of Science and Technology

Laboratory of mathematics and population dynamics

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

In this paper, we are interested in some theoretical and numerical studies of a special case of a degenerate nonlinear Schrödinger equation namely the...

Theoretical and numerical analysis of a degenerate nonlinear cubic Schrödinger equation

Published Online: May 28, 2022

Page range: 256 - 278

Received: Jul 28, 2021

Accepted: Apr 25, 2022

DOI: https://doi.org/10.2478/mjpaa-2022-0018

Keywords
Cubic Schrödinger equation, Time-dependent Schrödinger equation, Interior degeneracy, CrankNicolson method, Galerkin method

© 2022 Mohamed Alahyane et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Theoretical and numerical analysis of a degenerate nonlinear cubic Schrödinger equation

Published Online: May 28, 2022

Page range: 256 - 278

Received: Jul 28, 2021

Accepted: Apr 25, 2022

DOI: https://doi.org/10.2478/mjpaa-2022-0018

KeywordsCubic Schrödinger equation, Time-dependent Schrödinger equation, Interior degeneracy, CrankNicolson method, Galerkin method

© 2022 Mohamed Alahyane et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Cubic Schrödinger equation, Time-dependent Schrödinger equation, Interior degeneracy, CrankNicolson method, Galerkin method