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Stability Analysis, Modulation Instability, and Beta-Time Fractional Exact Soliton Solutions to the Van der Waals Equation

Author

Listed:
  • Haitham Qawaqneh

    (Department of Mathematics, Faculty of Science and Information Technology, Al-Zaytoonah University of Jordan, Amman 11733, Jordan)

  • Jalil Manafian

    (Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz 5166616471, Iran
    Natural Sciences Faculty, Lankaran State University, 50, H. Aslanov Str., Lankaran AZ 4200, Azerbaijan)

  • Mohammed Alharthi

    (Department of Mathematics, College of Sciences, University of Bisha, P.O. Box 344, Bisha 61922, Saudi Arabia)

  • Yasser Alrashedi

    (Department of Mathematics, College of Sciences, Taibah University, P.O. Box 344, Madinah 42353, Saudi Arabia)

Abstract

The study consists of the distinct types of the exact soliton solutions to an important model called the beta-time fractional (1 + 1)-dimensional non-linear Van der Waals equation. This model is used to explain the motion of molecules and materials. The Van der Waals equation explains the phase separation phenomenon. Noncovalent Van der Waals or dispersion forces usually have an effect on the structure, dynamics, stability, and function of molecules and materials in different branches of science, including biology, chemistry, materials science, and physics. Solutions are obtained, including dark, dark-singular, periodic wave, singular wave, and many more exact wave solutions by using the modified extended tanh function method. Using the fractional derivatives makes different solutions different from the existing solutions. The gained results will be of high importance in the interaction of quantum-mechanical fluctuations, granular matters, and other applications of the Van der Waals equation. The solutions may be useful in distinct fields of science and civil engineering, as well as some basic physical ones like those studied in geophysics. The results are verified and represented by two-dimensional, three-dimensional, and contour graphs by using Mathematica software. The obtained results are newer than the existing results. Stability analysis is also performed to check the stability of the concerned model. Furthermore, modulation instability is studied to study the stationary solutions of the concerned model. The results will be helpful in future studies of the concerned system. In the end, we can say that the method used is straightforward and dynamic, and it will be a useful tool for debating tough issues in a wide range of fields.

Suggested Citation

  • Haitham Qawaqneh & Jalil Manafian & Mohammed Alharthi & Yasser Alrashedi, 2024. "Stability Analysis, Modulation Instability, and Beta-Time Fractional Exact Soliton Solutions to the Van der Waals Equation," Mathematics, MDPI, vol. 12(14), pages 1-23, July.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:14:p:2257-:d:1438954
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    References listed on IDEAS

    as
    1. Tariq, Kalim U. & Wazwaz, Abdul-Majid & Javed, Rizwan, 2023. "Construction of different wave structures, stability analysis and modulation instability of the coupled nonlinear Drinfel’d–Sokolov–Wilson model," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    2. El-Wakil, S.A. & Abdou, M.A., 2007. "Modified extended tanh-function method for solving nonlinear partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 31(5), pages 1256-1264.
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