This paper presents a significant contribution in the form of a new general equation, namely the $ \mathfrak{q} $-deformed equation or the $ \mathfrak{q} $-deformed tanh-Gordon equation. The introduction of this novel equation opens up new possibilities for modeling physical systems that exhibit violated symmetries. By employing the $ (G'/G) $ expansion method, we have successfully derived solitary wave solutions for the newly defined $ \mathfrak{q} $-deformed equation under specific parameter regimes. These solutions provide valuable insights into the behavior of the system and its dynamics. To further validate the obtained analytical results, the numerical solution of the $ \mathfrak{q} $-deformed equation has been constructed by using the finite difference method. This numerical approach ensures the accuracy and reliability of the findings. To facilitate a comprehensive understanding of the results, we have included two- and three-dimensional tables and figures, which provide visual representations and comparisons between the analytical and numerical solutions. These graphical illustrations enhance the clarity and interpretation of the obtained data. The significance of the $ \mathfrak{q} $-deformation lies in its ability to model physical systems that exhibit deviations from standard symmetry properties, such as extensivity. This type of modeling is increasingly relevant in various fields, as it allows for a more accurate representation of real-world phenomena.
Citation: Khalid K. Ali, Weam G. Alharbi. Exploring unconventional optical soliton solutions for a novel $ \mathfrak{q} $-deformed mathematical model[J]. AIMS Mathematics, 2024, 9(6): 15202-15222. doi: 10.3934/math.2024738
This paper presents a significant contribution in the form of a new general equation, namely the $ \mathfrak{q} $-deformed equation or the $ \mathfrak{q} $-deformed tanh-Gordon equation. The introduction of this novel equation opens up new possibilities for modeling physical systems that exhibit violated symmetries. By employing the $ (G'/G) $ expansion method, we have successfully derived solitary wave solutions for the newly defined $ \mathfrak{q} $-deformed equation under specific parameter regimes. These solutions provide valuable insights into the behavior of the system and its dynamics. To further validate the obtained analytical results, the numerical solution of the $ \mathfrak{q} $-deformed equation has been constructed by using the finite difference method. This numerical approach ensures the accuracy and reliability of the findings. To facilitate a comprehensive understanding of the results, we have included two- and three-dimensional tables and figures, which provide visual representations and comparisons between the analytical and numerical solutions. These graphical illustrations enhance the clarity and interpretation of the obtained data. The significance of the $ \mathfrak{q} $-deformation lies in its ability to model physical systems that exhibit deviations from standard symmetry properties, such as extensivity. This type of modeling is increasingly relevant in various fields, as it allows for a more accurate representation of real-world phenomena.
| [1] |
H. Almusawa, K. K. Ali, A. M. Wazwaz, D. Baleanu, M. S. Osman, Protracted study on a real physical phenomenon generated by media inhomogeneities, Results Phys., 31 (2021), 104933. https://doi.org/10.1016/j.rinp.2021.104933 doi: 10.1016/j.rinp.2021.104933
|
| [2] |
O. Moaaz, H. Ramos, J. Awrejcewicz, Second-order Emden-Fowler neutral differential equations: A new precise criterion for oscillation, Appl. Math. Lett., 118 (2021), 107172. https://doi.org/10.1016/j.aml.2021.107172 doi: 10.1016/j.aml.2021.107172
|
| [3] |
O. Moaaz, C. Cesarano, A. Muhib, Some new oscillation results for fourth-order neutral differential equations, Eur. J. Pure Appl. Math., 13 (2020), 185–199. https://doi.org/10.29020/nybg.ejpam.v13i2.3654 doi: 10.29020/nybg.ejpam.v13i2.3654
|
| [4] |
A. Ahmad, R. Ali, I. Ahmad, F. A. Awwad, E. A. A. Ismail, Global stability of fractional order HIV/AIDS epidemic model under Caputo operator and its computational modeling, Fractal Fract., 7 (2023), 643. https://doi.org/10.3390/fractalfract7090643 doi: 10.3390/fractalfract7090643
|
| [5] |
R. Ali, A. S. Hendy, M. R. Ali, A. M. Hassan, F. A. Awwad, E. A. A. Ismail, Exploring propagating soliton solutions for the fractional Kudryashov-Sinelshchikov equation in a mixture of liquid-gas bubbles under the consideration of heat transfer and viscosity, Fractal Fract., 11 (2023), 773. https://doi.org/10.3390/fractalfract7110773 doi: 10.3390/fractalfract7110773
|
| [6] |
R. Ali, E. Tag-eldin, A comparative analysis of generalized and extended $(G'/G)$-expansion methods for travelling wave solutions of fractional Maccari's system with complex structure, Alex. Eng. J., 79 (2023), 508–530. https://doi.org/10.1016/j.aej.2023.08.007 doi: 10.1016/j.aej.2023.08.007
|
| [7] |
A. Arai, Exactly solvable supersymmetric quantum mechanics, J. Math. Anal. Appl., 158 (1991), 63–79. https://doi.org/10.1016/0022-247X(91)90267-4 doi: 10.1016/0022-247X(91)90267-4
|
| [8] |
A. Arai, Exact solutions of multi-component nonlinear Schrödinger and Klein-Gordon equations in two-dimensional space-time, J. Phys. A-Math. Gen., 34 (2001), 4281–4288. https://doi.org/10.1088/0305-4470/34/20/302 doi: 10.1088/0305-4470/34/20/302
|
| [9] |
B. J. Falaye, K. J. Oyewumi, M. Abbas, Exact solution of Schrödinger equation with q-deformed quantum potentials using Nikiforov-Uvarov method, Chinese Phys. B, 22 (2013), 110301. https://doi.org/10.1088/1674-1056/22/11/110301 doi: 10.1088/1674-1056/22/11/110301
|
| [10] |
A. Kurniawan, A. Suparmi, C. Cari, Approximate analytical solution of the Dirac equation with q-deformed hyperbolic Poschl-Teller potential and trigonometric Scarf Ⅱ non-central potential, Chinese Phys. B, 24 (2015), 030302. https://doi.org/10.1088/1674-1056/24/3/030302 doi: 10.1088/1674-1056/24/3/030302
|
| [11] |
Y. Shu, J. Chen, L. Chen, Bose-Einstein condensation of a q-deformed ideal Bose gas, Phys. Lett. A, 292 (2002), 309–314. https://doi.org/10.1016/S0375-9601(01)00816-7 doi: 10.1016/S0375-9601(01)00816-7
|
| [12] |
S. M. Ikhdair, Rotation and vibration of diatomic molecule in the spatially-dependent mass Schrödinger equation with generalized q-deformed Morse potential, Chem. Phys., 361 (2009), 9–17. https://doi.org/10.1016/j.chemphys.2009.04.023 doi: 10.1016/j.chemphys.2009.04.023
|
| [13] |
D. Bonatsos, E. N. Argyres, P. P. Raychev, SU-(1, 1) description of vibrational molecular spectra, J. Phys. A-Math. Gen., 24 (1991), 403–408. https://doi.org/10.1088/0305-4470/24/8/003 doi: 10.1088/0305-4470/24/8/003
|
| [14] |
H. Eleuch, Some analytical solitary wave solutions for the generalized q-deformed Sinh-Gordon equation $ \frac{\partial^2u}{\partial z \partial \mathcal{E}} = [\sinh_q (\beta u^\gamma)]^p -\Lambda$, Adv. Math. Phys., 2018 (2018), 5242757. http://dx.doi.org/10.1155/2018/5242757 doi: 10.1155/2018/5242757
|
| [15] |
K. K. Ali, Traveling wave solutions, numerical solutions, and stability analysis of the (2+1) conformal time-fractional generalized q-deformed sinh-Gordon equation, Nonlinear Eng., 13 (2024), 20220348. https://doi.org/10.1515/nleng-2022-0348 doi: 10.1515/nleng-2022-0348
|
| [16] |
K. K. Ali, A. Abdel-Aty, H. Eleuch, New soliton solutions for the conformal time derivative q-deformed physical model, Results Phys., 42 (2022), 105993. https://doi.org/10.1016/j.rinp.2022.105993 doi: 10.1016/j.rinp.2022.105993
|
| [17] |
K. K. Ali, N. Al-Harbi, A. Abdel-Aty, Traveling wave solutions to (3 + 1) conformal time derivative generalized q-deformed Sinh-Gordon equation, Alex. Eng. J., 2022 (2022), 1–12, https://doi.org/10.1016/j.aej.2022.10.020 doi: 10.1016/j.aej.2022.10.020
|
| [18] |
K. K. Ali, H. I. Alrebdi, N. A. M. Alsaif, A. Abdel-Aty, H. Eleuch, Analytical solutions for a new form of the generalized q-deformed Sinh-Gordon equation: $\frac{\partial^2u}{\partial z \partial \mathcal{E}} = e^{\alpha u}[\sinh_{\mathfrak{q}} (u^\gamma)]^p -\delta$, Symmetry, 15 (2023), 470. https://doi.org/10.3390/sym15020470 doi: 10.3390/sym15020470
|
| [19] |
M. Shallal, K. K. Ali, K. R. Raslan, H. Rezazadeh, A. Bekir, Exact solutions of the conformable fractional EW and MEW equations by a new generalized expansion method, J. Ocean Eng. Sci., 5 (2020), 323–329. https://doi.org/10.1016/j.joes.2019.12.004 doi: 10.1016/j.joes.2019.12.004
|
| [20] |
K. R. Raslan, K. K. Ali, Numerical study of MHD-duct flow using the two-dimensional finite difference method, Appl. Math. Inf. Sci., 14 (2020), 1–5. https://doi.org/10.18576/amis/140417 doi: 10.18576/amis/140417
|
| [21] | T. S. EL-Danaf, K. R. Raslan, K. K. Ali, New numerical treatment for the generalized regularized long wave equation based on finite difference scheme, Int. J. Soft Comput. Eng. (IJSCE), 4 (2014), 16–24. Available from: https://www.ijsce.org/wp-content/uploads/papers/v4i4/D2328094414.pdf. |
Figures(12) / Tables(6)
Khalid K. Ali, Weam G. Alharbi. Exploring unconventional optical soliton solutions for a novel $ \mathfrak{q} $-deformed mathematical model[J]. AIMS Mathematics, 2024, 9(6): 15202-15222. doi: 10.3934/math.2024738
DownLoad: