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A Mass- and Energy-Conserving Numerical Model for a Fractional Gross–Pitaevskii System in Multiple Dimensions

Author

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  • Adán J. Serna-Reyes

    (Centro de Ciencias Básicas, Universidad Autónoma de Aguascalientes, Aguascalientes 20100, Mexico)

  • Jorge E. Macías-Díaz

    (Department of Mathematics and Didactics of Mathematics, School of Digital Technologies, Tallinn University, 10120 Tallinn, Estonia
    Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Aguascalientes 20100, Mexico)

Abstract

This manuscript studies a double fractional extended p -dimensional coupled Gross–Pitaevskii-type system. This system consists of two parabolic partial differential equations with equal interaction constants, coupling terms, and spatial derivatives of the Riesz type. Associated with the mathematical model, there are energy and non-negative mass functions which are conserved throughout time. Motivated by this fact, we propose a finite-difference discretization of the double fractional Gross–Pitaevskii system which inherits the energy and mass conservation properties. As the continuous model, the mass is a non-negative constant and the solutions are bounded under suitable numerical parameter assumptions. We prove rigorously the existence of solutions for any set of initial conditions. As in the continuous system, the discretization has a discrete Hamiltonian associated. The method is implicit, multi-consistent, stable and quadratically convergent. Finally, we implemented the scheme computationally to confirm the validity of the mass and energy conservation properties, obtaining satisfactory results.

Suggested Citation

  • Adán J. Serna-Reyes & Jorge E. Macías-Díaz, 2021. "A Mass- and Energy-Conserving Numerical Model for a Fractional Gross–Pitaevskii System in Multiple Dimensions," Mathematics, MDPI, vol. 9(15), pages 1-31, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:15:p:1765-:d:601631
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    References listed on IDEAS

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    1. Wen Tan & Feng Ling Jiang & Chuang Xia Huang & Lan Zhou, 2012. "Synchronization for a Class of Fractional-Order Hyperchaotic System and Its Application," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-11, June.
    2. Manuel Duarte Ortigueira, 2006. "Riesz potential operators and inverses via fractional centred derivatives," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2006, pages 1-12, August.
    3. X. Wang & F. Liu & X. Chen, 2015. "Novel Second-Order Accurate Implicit Numerical Methods for the Riesz Space Distributed-Order Advection-Dispersion Equations," Advances in Mathematical Physics, Hindawi, vol. 2015, pages 1-14, November.
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