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Year 2025, Volume: 13 Issue: 3, 116 - 125, 06.09.2025

Mehmet Fatih Uçar

https://doi.org/10.36753/mathenot.1670727

Abstract

References

  • [1] Zakharov, V. E., Shabat, A. B.:Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Soviet Physics JETP. 34(1) 62–69 (1972).
  • [2] Sulem, C., Sulem, P.-L.:The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Springer (1999).
  • [3] Fibich, G.:The Nonlinear Schrödinger Equation: Singular Solutions and Optical Collapse. Springer (2015).
  • [4] Shukla, P. K., Eliasson, B.:Nonlinear aspects of quantum plasma physics. Physics-Uspekhi. 53(1), 51 (2010).
  • [5] Şenol, M., Gençyiğit, M., Koksal, M. E., Qureshi, S.:New analytical and numerical solutions to the (2+1)-dimensional conformable cpKP–BKP equation arising in fluid dynamics, plasma physics, and nonlinear optics. Optical and Quantum Electronics. 56(3), 352 (2023).
  • [6] Kivshar, Y. S., Agrawal, G. P.:Optical Solitons: From Fibers to Photonic Crystals. Academic Press (2003).
  • [7] Couairon, A., Mysyrowicz, A.:Femtosecond filamentation in transparent media. Physics Reports. 441(2-4), 47–189 (2007).
  • [8] Pitaevskii, L., Stringari, S.:Bose-Einstein Condensation. Oxford University Press (2003).
  • [9] Duan, W. S.:3+1 dimensional envelop waves and its stability in magnetized dusty plasma. Chaos, Solitons & Fractals. 27(4), 926–929 (2006).
  • [10] Haque, M. N., Mannan, A., Mamun, A. A.:The (3+1)-dimensional dust-acoustic waves in multi-components magnetoplasmas. Contributions to Plasma Physics. 59(10), e201900049 (2019).
  • [11] Lin, M., Zhang, X., Du, H., Duan, W.:Effect of nonthermal ions on (3+1)-dimensional envelope solitary wave in magnetized PLD dusty plasma. Indian Journal of Physics. 92(1), 129–136 (2018).
  • [12] Inan, I. E., Inc, M., Rezazadeh, H., Akinyemi, L.:Optical solitons of (3+1) dimensional and coupled nonlinear Schrodinger equations. Optical and Quantum Electronics. 54(4), 261 (2022).
  • [13] Ahmed, K. K., Ahmed, H. M., Rabie, W. B., Shehab, M. F.:Effect of noise on wave solitons for (3+1)-dimensional nonlinear Schrödinger equation in optical fiber. Indian Journal of Physics. 98(14), 4863–4882 (2024).
  • [14] Yang, N.:Exact wave patterns and chaotic dynamical behaviors of the extended (3+1)-dimensional NLSE. AIMS Mathematics. 9(11), 31274–31294 (2024).
  • [15] Nisar, K. S., Inan, I. E., Inc, M., Rezazadeh, H.:Properties of some higher-dimensional nonlinear Schrodinger equations. Results in Physics. 31(Dec 2021), 105073 (2021).
  • [16] Liang, J., Liu, H., Liu, F., Yi, L.:Analytical solutions to the (3+1)-dimensional generalized nonlinear Schrodinger equation with varying parameters. Journal of Physics A: Mathematical and Theoretical. 42(33), 335204 (2009).
  • [17] Kumar, S., Kukkar, A.:Dynamics of several optical soliton solutions of a (3+1)-dimensional nonlinear Schrödinger equation with parabolic law in optical fibers. Modern Physics Letters B. 39(9) (2025).
  • [18] Mathanaranjan, T.:Optical solitons and stability analysis for the new (3+1)-dimensional nonlinear Schrödinger equation. Journal of Nonlinear Optical Physics & Materials. 32(2), 2350016 (2023).
  • [19] Liao, B.,Wang, Z.:Modified three-dimensional nonlinear Schrödinger equation in finite water depth for gravity waves with influence of slowly varying currents. Physics of Fluids. 36(10), 102104 (2024).
  • [20] Wang, G.:A new (3 + 1)-dimensional Schrödinger equation: derivation, soliton solutions and conservation laws. Nonlinear Dynamics. 104(2), 1595–1602 (2021).
  • [21] Wang, G., Wang, X., Guan, F., Song, H.:Exact solutions of an extended (3+1)-dimensional nonlinear Schrödinger equation with cubic-quintic nonlinearity term. Optik. 279(May 2023), 170768 (2023).
  • [22] Mirzazadeh, M., Hashemi, M. S., Akbulu, A., Ur Rehman, H., Iqbal, I., Eslami, M.:Dynamics of optical solitons in the extended (3+1)-dimensional nonlinear conformable Kudryashov equation with generalized anti-cubic nonlinearity. Mathematical Methods in the Applied Sciences. 47(7), 5355–5375 (2024).
  • [23] Ozisik, M.:Novel (2+1) and (3+1) forms of the Biswas–Milovic equation and optical soliton solutions via two efficient techniques. Optik. 269(Nov 2022) , 169798 (2022).
  • [24] Murad, M. A. S.:Analyzing the time-fractional (3+1)-dimensional nonlinear Schrödinger equation: a new Kudryashov approach and optical solutions. International Journal of Computer Mathematics. 101(5), 524–537 (2024).
  • [25] Sirisubtawee, S., Koonprasert, S., Sungnul, S.:New exact solutions of the conformable space-time Sharma–Tasso–Olver equation using two reliable methods. Symmetry. 12(4), 644 (2020).
  • [26] Al-Deiakeh, R., Alquran, M., Ali, M., Qureshi, S., Momani, S., Malkawi, A. A. R.:Lie symmetry, convergence analysis, explicit solutions, and conservation laws for the time-fractional modified Benjamin-Bona-Mahony equation. Journal of Applied Mathematics and Computational Mechanics. 23(1), 19–31 (2024).
  • [27] Shehab, M. F., El-Sheikh, M. M. A., Ahmed, H. M., El-Gaber, A. A., Alkhatib, S.:Effects of Wiener process on analytical wave solutions for (3+1) dimensional nonlinear Schrödinger equation using modified extended mapping method. Results in Physics. 56(Jan 2024), 107297 (2024).
  • [28] Eslami, M., Rezazadeh, H.:The first integral method for Wu–Zhang system with conformable time-fractional derivative. Calcolo. 53(1) , 475–485 (2016).
  • [29] Wu, H.-Y., Jiang, L.-H.:Vortex soliton solutions of a (3+1)-dimensional Gross-Pitaevskii equation with partially nonlocal distributed coefficients under a linear potential. Nonlinear Dynamics. 101(4), 2441–2448 (2020).
  • [30] Lai, X.-J., Cai, X.-O.:Chirped Wave Solutions of a Generalized (3+1)-Dimensional Nonlinear Schrodinger Equation. Zeitschrift für Naturforschung A. 66(6), 392–400 (2011).
  • [31] Yang, J.:Nonlinear Waves in Integrable and Nonintegrable Systems. SIAM (2010).
  • [32] Alquran, M., Ali, M., Gharaibeh, F., Qureshi, S.:Novel investigations of dual-wave solutions to the Kadomtsev–Petviashvili model involving second-order temporal and spatial–temporal dispersion terms. Partial Differential Equations in Applied Mathematics. 8(Dec 2023), 100543 (2023).
  • [33] Wazwaz, A.-M., Mehanna, M.:Higher-order Sasa–Satsuma equation: Bright and dark optical solitons. Optik. 243(Oct 2021), 167421 (2021).
  • [34] Wang, K.-J., Wang, G.-D., Shi, F.:Nonlinear dynamics of soliton molecules, hybrid interactions and other wave solutions for the (3+1)-dimensional generalized Kadomtsev-Petviashvili-Boussinesq equation. Modern Physics Letters B. 38 (25), 254194 (2024).
  • [35] Ozisik, M., Secer, A., Bayram, M.:Soliton Waves with the (3+1)-Dimensional Kadomtsev–Petviashvili–Boussinesq Equation in Water Wave Dynamics. Symmetry. 15(1), 165 (2023).
  • [36] Inc, M., Aliyu, A. I., Yusuf, A., Baleanu, D.:Optical solitons and modulation instability analysis with (3+1)-dimensional nonlinear Schrodinger equation. Superlattices and Microstructures. 112(Dec 2017), 296–302 (2017).
  • [37] Ozisik, M., Secer, A., Bayram, M.:(3+1)-dimensional Sasa–Satsuma equation under the effect of group velocity dispersion, self-frequency shift and self-steepening. Optik. 275 (March 2023), 170609 (2023).
  • [38] Korpinar, Z., Tchier, F., Inc, M.:On Optical Solitons of the Fractional (3+1)-Dimensional NLSE With Conformable Derivatives. Frontiers in Physics. 8(Apr 2020), 87 (2020).
  • [39] Korpinar, Z., Inc, M., Almohsen, B., Bayram, M.:Optical solitons for the fractional (3+1)-dimensional NLSE with power law nonlinearities by using conformable derivatives. Indian Journal of Physics. 95(10), 2143–2154 (2021).
  • [40] Kudryashov, N. A.:Method for finding highly dispersive optical solitons of nonlinear differential equations. Optik. 206(March 2020), 163550 (2020).
  • [41] Ozisik, M., Secer, A., Bayram, M., Aydin, H.:An encyclopedia of Kudryashov’s integrability approaches applicable to optoelectronic devices. Optik. 265(Sep 2022), 169499 (2022).

Optical Soliton Solutions of a $(3+1)$-Dimensional Nonlinear Schrödinger Equation with Cubic-Quintic-Septic Nonlinearity

Year 2025, Volume: 13 Issue: 3, 116 - 125, 06.09.2025

Mehmet Fatih Uçar

https://doi.org/10.36753/mathenot.1670727

Abstract

In this paper, we studied the extended (3+1)-dimensional NLSE that has cubic-quintic-septic nonlinearity term. To obtain meaningful results, we first defined the wave transform and applied it to the NLSE. We then derived the ordinary differential equation (ODE). We gave brief information about the new Kudryashov method (nKM) and applied it to our problem. Finally, we solved the system using Maple. Using these solutions, we obtained a bright soliton and then illustrated the graphics and analyzed the effect of the parameters of the equation on the graph of such a soliton solution.

Keywords

Optical soliton Parameter effect The new Kudryashov method (3+1)-dimensional NLSE

References

  • [1] Zakharov, V. E., Shabat, A. B.:Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Soviet Physics JETP. 34(1) 62–69 (1972).
  • [2] Sulem, C., Sulem, P.-L.:The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Springer (1999).
  • [3] Fibich, G.:The Nonlinear Schrödinger Equation: Singular Solutions and Optical Collapse. Springer (2015).
  • [4] Shukla, P. K., Eliasson, B.:Nonlinear aspects of quantum plasma physics. Physics-Uspekhi. 53(1), 51 (2010).
  • [5] Şenol, M., Gençyiğit, M., Koksal, M. E., Qureshi, S.:New analytical and numerical solutions to the (2+1)-dimensional conformable cpKP–BKP equation arising in fluid dynamics, plasma physics, and nonlinear optics. Optical and Quantum Electronics. 56(3), 352 (2023).
  • [6] Kivshar, Y. S., Agrawal, G. P.:Optical Solitons: From Fibers to Photonic Crystals. Academic Press (2003).
  • [7] Couairon, A., Mysyrowicz, A.:Femtosecond filamentation in transparent media. Physics Reports. 441(2-4), 47–189 (2007).
  • [8] Pitaevskii, L., Stringari, S.:Bose-Einstein Condensation. Oxford University Press (2003).
  • [9] Duan, W. S.:3+1 dimensional envelop waves and its stability in magnetized dusty plasma. Chaos, Solitons & Fractals. 27(4), 926–929 (2006).
  • [10] Haque, M. N., Mannan, A., Mamun, A. A.:The (3+1)-dimensional dust-acoustic waves in multi-components magnetoplasmas. Contributions to Plasma Physics. 59(10), e201900049 (2019).
  • [11] Lin, M., Zhang, X., Du, H., Duan, W.:Effect of nonthermal ions on (3+1)-dimensional envelope solitary wave in magnetized PLD dusty plasma. Indian Journal of Physics. 92(1), 129–136 (2018).
  • [12] Inan, I. E., Inc, M., Rezazadeh, H., Akinyemi, L.:Optical solitons of (3+1) dimensional and coupled nonlinear Schrodinger equations. Optical and Quantum Electronics. 54(4), 261 (2022).
  • [13] Ahmed, K. K., Ahmed, H. M., Rabie, W. B., Shehab, M. F.:Effect of noise on wave solitons for (3+1)-dimensional nonlinear Schrödinger equation in optical fiber. Indian Journal of Physics. 98(14), 4863–4882 (2024).
  • [14] Yang, N.:Exact wave patterns and chaotic dynamical behaviors of the extended (3+1)-dimensional NLSE. AIMS Mathematics. 9(11), 31274–31294 (2024).
  • [15] Nisar, K. S., Inan, I. E., Inc, M., Rezazadeh, H.:Properties of some higher-dimensional nonlinear Schrodinger equations. Results in Physics. 31(Dec 2021), 105073 (2021).
  • [16] Liang, J., Liu, H., Liu, F., Yi, L.:Analytical solutions to the (3+1)-dimensional generalized nonlinear Schrodinger equation with varying parameters. Journal of Physics A: Mathematical and Theoretical. 42(33), 335204 (2009).
  • [17] Kumar, S., Kukkar, A.:Dynamics of several optical soliton solutions of a (3+1)-dimensional nonlinear Schrödinger equation with parabolic law in optical fibers. Modern Physics Letters B. 39(9) (2025).
  • [18] Mathanaranjan, T.:Optical solitons and stability analysis for the new (3+1)-dimensional nonlinear Schrödinger equation. Journal of Nonlinear Optical Physics & Materials. 32(2), 2350016 (2023).
  • [19] Liao, B.,Wang, Z.:Modified three-dimensional nonlinear Schrödinger equation in finite water depth for gravity waves with influence of slowly varying currents. Physics of Fluids. 36(10), 102104 (2024).
  • [20] Wang, G.:A new (3 + 1)-dimensional Schrödinger equation: derivation, soliton solutions and conservation laws. Nonlinear Dynamics. 104(2), 1595–1602 (2021).
  • [21] Wang, G., Wang, X., Guan, F., Song, H.:Exact solutions of an extended (3+1)-dimensional nonlinear Schrödinger equation with cubic-quintic nonlinearity term. Optik. 279(May 2023), 170768 (2023).
  • [22] Mirzazadeh, M., Hashemi, M. S., Akbulu, A., Ur Rehman, H., Iqbal, I., Eslami, M.:Dynamics of optical solitons in the extended (3+1)-dimensional nonlinear conformable Kudryashov equation with generalized anti-cubic nonlinearity. Mathematical Methods in the Applied Sciences. 47(7), 5355–5375 (2024).
  • [23] Ozisik, M.:Novel (2+1) and (3+1) forms of the Biswas–Milovic equation and optical soliton solutions via two efficient techniques. Optik. 269(Nov 2022) , 169798 (2022).
  • [24] Murad, M. A. S.:Analyzing the time-fractional (3+1)-dimensional nonlinear Schrödinger equation: a new Kudryashov approach and optical solutions. International Journal of Computer Mathematics. 101(5), 524–537 (2024).
  • [25] Sirisubtawee, S., Koonprasert, S., Sungnul, S.:New exact solutions of the conformable space-time Sharma–Tasso–Olver equation using two reliable methods. Symmetry. 12(4), 644 (2020).
  • [26] Al-Deiakeh, R., Alquran, M., Ali, M., Qureshi, S., Momani, S., Malkawi, A. A. R.:Lie symmetry, convergence analysis, explicit solutions, and conservation laws for the time-fractional modified Benjamin-Bona-Mahony equation. Journal of Applied Mathematics and Computational Mechanics. 23(1), 19–31 (2024).
  • [27] Shehab, M. F., El-Sheikh, M. M. A., Ahmed, H. M., El-Gaber, A. A., Alkhatib, S.:Effects of Wiener process on analytical wave solutions for (3+1) dimensional nonlinear Schrödinger equation using modified extended mapping method. Results in Physics. 56(Jan 2024), 107297 (2024).
  • [28] Eslami, M., Rezazadeh, H.:The first integral method for Wu–Zhang system with conformable time-fractional derivative. Calcolo. 53(1) , 475–485 (2016).
  • [29] Wu, H.-Y., Jiang, L.-H.:Vortex soliton solutions of a (3+1)-dimensional Gross-Pitaevskii equation with partially nonlocal distributed coefficients under a linear potential. Nonlinear Dynamics. 101(4), 2441–2448 (2020).
  • [30] Lai, X.-J., Cai, X.-O.:Chirped Wave Solutions of a Generalized (3+1)-Dimensional Nonlinear Schrodinger Equation. Zeitschrift für Naturforschung A. 66(6), 392–400 (2011).
  • [31] Yang, J.:Nonlinear Waves in Integrable and Nonintegrable Systems. SIAM (2010).
  • [32] Alquran, M., Ali, M., Gharaibeh, F., Qureshi, S.:Novel investigations of dual-wave solutions to the Kadomtsev–Petviashvili model involving second-order temporal and spatial–temporal dispersion terms. Partial Differential Equations in Applied Mathematics. 8(Dec 2023), 100543 (2023).
  • [33] Wazwaz, A.-M., Mehanna, M.:Higher-order Sasa–Satsuma equation: Bright and dark optical solitons. Optik. 243(Oct 2021), 167421 (2021).
  • [34] Wang, K.-J., Wang, G.-D., Shi, F.:Nonlinear dynamics of soliton molecules, hybrid interactions and other wave solutions for the (3+1)-dimensional generalized Kadomtsev-Petviashvili-Boussinesq equation. Modern Physics Letters B. 38 (25), 254194 (2024).
  • [35] Ozisik, M., Secer, A., Bayram, M.:Soliton Waves with the (3+1)-Dimensional Kadomtsev–Petviashvili–Boussinesq Equation in Water Wave Dynamics. Symmetry. 15(1), 165 (2023).
  • [36] Inc, M., Aliyu, A. I., Yusuf, A., Baleanu, D.:Optical solitons and modulation instability analysis with (3+1)-dimensional nonlinear Schrodinger equation. Superlattices and Microstructures. 112(Dec 2017), 296–302 (2017).
  • [37] Ozisik, M., Secer, A., Bayram, M.:(3+1)-dimensional Sasa–Satsuma equation under the effect of group velocity dispersion, self-frequency shift and self-steepening. Optik. 275 (March 2023), 170609 (2023).
  • [38] Korpinar, Z., Tchier, F., Inc, M.:On Optical Solitons of the Fractional (3+1)-Dimensional NLSE With Conformable Derivatives. Frontiers in Physics. 8(Apr 2020), 87 (2020).
  • [39] Korpinar, Z., Inc, M., Almohsen, B., Bayram, M.:Optical solitons for the fractional (3+1)-dimensional NLSE with power law nonlinearities by using conformable derivatives. Indian Journal of Physics. 95(10), 2143–2154 (2021).
  • [40] Kudryashov, N. A.:Method for finding highly dispersive optical solitons of nonlinear differential equations. Optik. 206(March 2020), 163550 (2020).
  • [41] Ozisik, M., Secer, A., Bayram, M., Aydin, H.:An encyclopedia of Kudryashov’s integrability approaches applicable to optoelectronic devices. Optik. 265(Sep 2022), 169499 (2022).
There are 41 citations in total.

Details

Primary Language English
Subjects Mathematical Methods and Special Functions, Applied Mathematics (Other)
Journal Section Articles
Authors

Mehmet Fatih Uçar 0000-0002-5542-2222

Early Pub Date July 30, 2025
Publication Date September 6, 2025
Submission Date April 6, 2025
Acceptance Date July 1, 2025
Published in Issue Year 2025 Volume: 13 Issue: 3

Cite

APA Uçar, M. F. (2025). Optical Soliton Solutions of a $(3+1)$-Dimensional Nonlinear Schrödinger Equation with Cubic-Quintic-Septic Nonlinearity. Mathematical Sciences and Applications E-Notes, 13(3), 116-125. https://doi.org/10.36753/mathenot.1670727
AMA Uçar MF. Optical Soliton Solutions of a $(3+1)$-Dimensional Nonlinear Schrödinger Equation with Cubic-Quintic-Septic Nonlinearity. Math. Sci. Appl. E-Notes. September 2025;13(3):116-125. doi:10.36753/mathenot.1670727
Chicago Uçar, Mehmet Fatih. “Optical Soliton Solutions of a $(3+1)$-Dimensional Nonlinear Schrödinger Equation With Cubic-Quintic-Septic Nonlinearity”. Mathematical Sciences and Applications E-Notes 13, no. 3 (September 2025): 116-25. https://doi.org/10.36753/mathenot.1670727.
EndNote Uçar MF (September 1, 2025) Optical Soliton Solutions of a $(3+1)$-Dimensional Nonlinear Schrödinger Equation with Cubic-Quintic-Septic Nonlinearity. Mathematical Sciences and Applications E-Notes 13 3 116–125.
IEEE M. F. Uçar, “Optical Soliton Solutions of a $(3+1)$-Dimensional Nonlinear Schrödinger Equation with Cubic-Quintic-Septic Nonlinearity”, Math. Sci. Appl. E-Notes, vol. 13, no. 3, pp. 116–125, 2025, doi: 10.36753/mathenot.1670727.
ISNAD Uçar, Mehmet Fatih. “Optical Soliton Solutions of a $(3+1)$-Dimensional Nonlinear Schrödinger Equation With Cubic-Quintic-Septic Nonlinearity”. Mathematical Sciences and Applications E-Notes 13/3 (September2025), 116-125. https://doi.org/10.36753/mathenot.1670727.
JAMA Uçar MF. Optical Soliton Solutions of a $(3+1)$-Dimensional Nonlinear Schrödinger Equation with Cubic-Quintic-Septic Nonlinearity. Math. Sci. Appl. E-Notes. 2025;13:116–125.
MLA Uçar, Mehmet Fatih. “Optical Soliton Solutions of a $(3+1)$-Dimensional Nonlinear Schrödinger Equation With Cubic-Quintic-Septic Nonlinearity”. Mathematical Sciences and Applications E-Notes, vol. 13, no. 3, 2025, pp. 116-25, doi:10.36753/mathenot.1670727.
Vancouver Uçar MF. Optical Soliton Solutions of a $(3+1)$-Dimensional Nonlinear Schrödinger Equation with Cubic-Quintic-Septic Nonlinearity. Math. Sci. Appl. E-Notes. 2025;13(3):116-25.
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