AccScience Publishing / IJOCTA / Online First / DOI: 10.36922/IJOCTA026330212
Submit to IJOCTA
Cite this article
7
Download
125
Views
Related Info Links
Google Scholar
More by Authors Links
Sekson Sirisubtawee
Journal Browser
Volume | Year
Issue
Search
News and Announcements
View All
RESEARCH ARTICLE

Dynamical features and dispersive soliton solutions of a fractional extended Schrödinger model incorporating cubic-quadratic nonlinear terms

Shafqat Ur Rehman1,2 ,  Sekson Sirisubtawee1,3* ,  Khomsan Neamprem1,3 ,  Chaiyod Kamthorncharoen1,3
Show Less
1 Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok , Thailand
2 Department of Mathematics, Grand Asian University, Sialkot , Pakistan
3 Centre of Excellence in Mathematics, MHESI, Bangkok , Thailand
IJOCTA, 026330212 https://doi.org/10.36922/IJOCTA026330212
Received: 13 August 2026 | Revised: 11 September 2026 | Accepted: 18 September 2026 | Published online: 12 October 2026
© 2026 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
Download PDF
XML
Cite
Abstract

This study is devoted to investigating soliton solutions and qualitative analysis of the M-fractional extended nonlinear Schrödinger (NLS) equation with cubicquadratic nonlinear terms. This research is motivated by the significance of the extended NLS model in optics, quantum mechanics, plasma physics, electromagnetic wave propagation, and the approximate description of dense quasione- dimensional Bose–Einstein condensates (BECs) involving repulsive contact interactions and long-range dipole-dipole attraction. The research gap concerns the limited understanding of the combined solitonic, modulational, and dynamical characteristics of this fractional cubic-quadratic model. The novelty lies in integrating the extended rational sine-cosine/sinh-cosh (ERSC–SHC) and generalized exponential rational function (GERFM) methods with modulation instability (MI), sensitivity analysis, bifurcation analysis, and chaos analysis. The contributions include bright, dark, singular, periodic, and exponential soliton structures in single and combined forms, MI regions obtained through linear stability theory, and qualitative analysis supported by graphical representations and phase portraits. A comparative analysis demonstrates distinct dynamical features, providing a foundation for investigations of nonlinear wave propagation and BEC dynamics.

Keywords
Soliton solutions
M-truncated extended NLS equation
ERSC–SHC method
GERFM
Modulation instability
Sensitivity analysis
Bifurcation and chaos
Funding
This research was funded by King Mongkut's University of Technology North Bangkok, Contract No. KMUTNB-69-KNOW-47. In addition, the first author was financially supported by King Mongkut's University of Technology North Bangkok under contract no. KMUTNB-Post-69-02.
Conflict of interest
The authors declare they have no competing interests.
References
  1. Mohan B, Kumar S. Generalization and analytic exploration of soliton solutions for nonlinear evolution equations via a novel symbolic approach in fluids and nonlinear sciences. Chin J Phys. 2024;92:10-21. https://doi.org/10.1016/j.cjph.2024.09.004
  2. Sagib M, Saha BK, Mohanty SK, Rahman MS. Diverse soliton solutions to the nonlinear partial differential equations related to electrical transmission line. Phys Scr. 2024;99(9):095219. https://doi.org/10.1088/1402-4896/ad69e6
  3. Wang KL. New dynamical behaviors and soliton solutions of the coupled nonlinear Schrödinger equation. Int J Geom Methods Mod Phys. 2025;22(8):2550047. https://doi.org/10.1142/S0219887825500471
  4. Bilal M, Rehman SU, Alawaideh YM, Popa IL, Al-khamiseh BM. On the study of multiple wave profiles and hybrid phenomena to the newly developed multi-component Gross-Pitaevskii system and stability analysis. Ain Shams Eng J. 2025;16(9):103510. https://doi.org/10.1016/j.asej.2025.103510
  5. Khan MAU, Sadaf M, Akram G, Birhanu A, Rehan K, Hamed YS. Soliton solutions of nonlinear coupled Davey-Stewartson Fokas system using modified auxiliary equation method and extended (G′/G²)-expansion method. Sci Rep. 2024;14(1):21949. https://doi.org/10.1038/s41598-024-72571-0
  6. Rehman SU, Ahmad J. Stability analysis and novel optical pulses to Kundu-Mukherjee-Naskar model in birefringent fibers. Int J Mod Phys B. 2024;38(15):2450192. https://doi.org/10.1142/S0217979224501923
  7. Sulaiman TA, Bulut H. The new extended rational SGEEM for construction of optical solitons to the (2+1)-dimensional Kundu-Mukherjee-Naskar model. Appl Math Nonlinear Sci. 2019;4(2):513-522. https://doi.org/10.2478/AMNS.2019.2.00048
  8. Zhang J, Manafian J, Raut S, Roy S, Mahmoud KH, Alsubaie ASA. Study of two soliton and shock wave structures by weighted residual method and Hirota bilinear approach. Nonlinear Dyn. 2024;112(14):12375-12391. https://doi.org/10.1007/s11071-024-09706-5
  9. Tipu GH, Faridi WA, Myrzakulova Z, Myrzakulov R, AlQahtani SA, AlQahtani NF, et al. On optical soliton wave solutions of non-linear Kairat-X equation via new extended direct algebraic method. Opt Quantum Electron. 2024;56(4):655. https://doi.org/10.1007/s11082-024-06369-9
  10. Wang SF. Dromion solutions of nonlinear BKK equations using the improved F-expansion method. Pramana J Phys. 2025;99(2):60. https://doi.org/10.1007/s12043-025-02915-6
  11. Batool S, Arshad M, Perveen N, Sarwar S. Bright optical solution for fractional Lakshmanan-Porsezian-Daniel with spatio temporal dispersion by improved Adomian decomposition method. Opt Quantum Electron. 2024;56(7):1137. https://doi.org/10.1007/s11082-024-07048-5
  12. Zayed EME, Alurrfi KAE. The (G′/G, 1/G)-expansion method and its applications to two nonlinear Schrödinger equations describing the propagation of femtosecond pulses in nonlinear optical fibers. Optik. 2016;127(4):1581-1589. https://doi.org/10.1016/j.ijleo.2015.11.027
  13. Tahir M, Awan AU, Abro KA. Extraction of optical solitons in birefringent fibers for Biswas-Arshed equation via extended trial equation method. Nonlinear Eng. 2021;10(1):146-158. https://doi.org/10.1515/nleng-2021-0011
  14. Ozisik M, Secer A, Bayram M. On the investigation of chiral solitons via modified new Kudryashov method. Int J Geom Methods Mod Phys. 2023;20(07):2350117. https://doi.org/10.1142/S0219887823501177
  15. Sabi'u J, Ibrahim IS, Neamprem K, Sungnul S, Sirisubtawee S. Generalized modified unstable nonlinear Schrödinger's equation: Optical solitons and modulation instability. Mathematics. 2025;13(12):2032. https://doi.org/10.3390/math13122032
  16. Bilal M, Ren J, Alsubaie ASA, Mahmoud KH, Inc M. Dynamics of nonlinear diverse wave propagation to improved Boussinesq model in weakly dispersive medium of shallow waters or ion acoustic waves using efficient technique. Opt Quantum Electron. 2024;56(1):21. https://doi.org/10.1007/s11082-023-05587-x
  17. Hussain E, Malik S, Yadav A, Shah SAA, Iqbal MAB, Ragab AE, et al. Qualitative analysis and soliton solutions of nonlinear extended quantum Zakharov-Kuznetsov equation. Nonlinear Dyn. 2024;112(21):19295-19310. https://doi.org/10.1007/s11071-024-09992-z
  18. Younis M, Bilal M, Rehman SU, Seadawy AR, Rizvi STR. Perturbed optical solitons with conformable time-space fractional Gerdjikov-Ivanov equation. Math Sci. 2022;16(4):431-443. https://doi.org/10.1007/s40096-021-00431-3
  19. Younas T, Ahmad J. Novel soliton insights into generalized fractional Tzitzéica-type evolution equations using the modified Khater method. Mod Phys Lett B. 2025;39(10):2450441. https://doi.org/10.1142/S0217984924504414
  20. Lan ZZ. Multi-soliton solutions, breather-like and bound-state solitons for complex modified Korteweg-de Vries equation in optical fibers. Chin Phys B. 2024;33(6):060201. https://doi.org/10.1088/1674-1056/ad39d7
  21. Kumar S, Kukkar A. Dynamics of several optical soliton solutions of a (3+1)-dimensional nonlinear Schrödinger equation with parabolic law in optical fibers. Mod Phys Lett B. 2025;39(09):2450453. https://doi.org/10.1142/S0217984924504530
  22. Asghari Y, Eslami M, Matinfar M, Rezazadeh H. Novel soliton solution of discrete nonlinear Schrödinger system in nonlinear optical fiber. Alex Eng J. 2024;90:7-16. https://doi.org/10.1016/j.aej.2024.01.024
  23. Akram S, Ahmad J, Rehman SU. Stability analysis and dynamical behavior of solitons in nonlinear optics modelled by Lakshmanan-Porsezian-Daniel equation. Opt Quantum Electron. 2023;55(8):685. https://doi.org/10.1007/s11082-023-04986-4
  24. Das A, Karmakar B, Biswas A, Yıldırım Y, Alghamdi AA. Chirped periodic waves and solitary waves for a generalized derivative resonant nonlinear Schrödinger equation with cubic-quintic nonlinearity. Nonlinear Dyn. 2023;111(16):15347-15371. https://doi.org/10.1007/s11071-023-08640-2
  25. Abdullaev FK, Gammal A, Tomio L, Frederico T. Stability of trapped Bose-Einstein condensates. Phys Rev A. 2001;63(4):043604. https://doi.org/10.1103/PhysRevA.63.043604
  26. Benney DJ, Newell AC. The propagation of nonlinear wave envelopes. J Math Phys. 1967;46(1-4):133-139. https://doi.org/10.1002/sapm1967461133
  27. Triki H, Biswas A, Moshokoa SP, Belic M. Optical solitons and conservation laws with quadratic-cubic nonlinearity. Optik. 2017;128:63-70. https://doi.org/10.1016/j.ijleo.2016.10.010
  28. Pal R, Loomba S, Kumar CN. Chirped self-similar waves for quadratic-cubic nonlinear Schrödinger equation. Ann Phys. 2017;387:213-221. https://doi.org/10.1016/j.aop.2017.10.007
  29. Fujioka J, Cortés E, Pérez-Pascual R, Rodríguez RF, Espinosa A, Malomed BA. Chaotic solitons in the quadratic-cubic nonlinear Schrödinger equation under nonlinearity management. Chaos. 2011;21(3):033120. https://doi.org/10.1063/1.3629985
  30. Biswas A, Ullah MZ, Asma M, Zhou Q, Moshokoa SP, Belic M. Optical solitons with quadratic-cubic nonlinearity by semi-inverse variational principle. Optik. 2017;139:16-19. https://doi.org/10.1016/j.ijleo.2017.03.111
  31. Aslan EC, Inc M. Soliton solutions of NLSE with quadratic-cubic nonlinearity and stability analysis. Waves Random Complex Media. 2017;27(4):594-601. https://doi.org/10.1080/17455030.2017.1286060
  32. Inc M, Aliyu AI, Yusuf A, Baleanu D. Optical solitons and modulation instability analysis to the quadratic-cubic nonlinear Schrödinger equation. Nonlinear Anal Model Control. 2019;24(1):20-33. https://doi.org/10.15388/NA.2019.1.2
  33. Attia RA, Khater M, El-Sayed Ahmed A, El-Shorbagy MA. Accurate sets of solitary solutions for the quadratic-cubic fractional nonlinear Schrödinger equation. AIP Adv. 2021;11(5):055105. https://doi.org/10.1063/5.0050624
  34. Jiang X, Chen X, Chi M, Chen J. On Hopf bifurcation and control for a delay systems. Appl Math Comput. 2020;370:124906. https://doi.org/10.1016/j.amc.2019.124906
  35. Lin J, Xu C, Zhao Y, Pang Y, Liu Z, Shen J. Bifurcation and control of a predator-prey system with two time delays. J Appl Anal Comput. 2026;16(2):807-859. https://doi.org/10.11948/20250164
  36. Cinar M, Onder I, Secer A, Yusuf A, Sulaiman TA, Bayram M, et al. Soliton solutions of (2+1) dimensional Heisenberg ferromagnetic spin equation by the extended rational sine-cosine and sinh-cosh method. Int J Appl Comput Math. 2021;7(4):135. https://doi.org/10.1007/s40819-021-01076-5
  37. Hamza AE, Suhail M, Alsulami A, Mustafa A, Aldwoah K, Saber H. Exploring soliton solutions and chaotic dynamics in the (3+1)-dimensional Wazwaz-Benjamin-Bona-Mahony equation: A generalized rational exponential function approach. Fractal Fract. 2024;8(10):592. https://doi.org/10.3390/fractalfract8100592
  38. Farooq K, Tedjani AH, Li Z, Hussain E. Soliton dynamics of the nonlinear Kodama equation with M-truncated derivative via two innovative schemes: The generalized Arnous method and the Kudryashov method. Fractal Fract. 2025;9(7):436. https://doi.org/10.3390/fractalfract9070436
  39. Ehsan H, Abbas M, Nazir T, Mohammed PO, Chorfi N, Baleanu D. Efficient analytical algorithms to study Fokas dynamical models involving M-truncated derivative. Qual Theory Dyn Syst. 2024;23(1):49. https://doi.org/10.1007/s12346-023-00890-0
  40. Gu M, Liu F, Li J, Peng C, Li Z. Explicit solutions of the generalized Kudryashov's equation with truncated M-fractional derivative. Sci Rep. 2024;14(1):21714. https://doi.org/10.1038/s41598-024-72610-w
  41. Li Z. Optical solutions of the nonlinear Kodama equation with the M-truncated derivative via the extended (G′/G)-expansion method. Fractal Fract. 2025;9(5):300. https://doi.org/10.3390/fractalfract9050300
  42. Ismael HF, Baskonus HM, Bulut H, Gao W. Instability modulation and novel optical soliton solutions to the Gerdjikov-Ivanov equation with M-fractional. Opt Quantum Electron. 2023;55(4):303. https://doi.org/10.1007/s11082-023-04581-7
  43. Shakeel M, Alshammari FS, Ahmadzai HG. Optical soliton solutions, dynamical and sensitivity analysis for fractional perturbed Gerdjikov-Ivanov equation. Sci Rep. 2025;15(1):30843. https://doi.org/10.1038/s41598-025-09571-1
  44. Rehman SU, Bilal M, Ren J, Bayram M, Inc M. Sensitivity analysis and dynamics of optical dromions in conformable generalized nonlinear Schrödinger systems. Phys Lett A. 2025;531:130168. https://doi.org/10.1016/j.physleta.2024.130168
  45. Shahen NHM, Amin MA, Foyjonnesa, Rahman MM. Dynamics of computational waveform: A study of bifurcation, chaos, and sensitivity analysis. PLoS One. 2025;20(7):e0326230. https://doi.org/10.1371/journal.pone.0326230
  46. Li P, Shi S, Xu C, Rahman MU. Bifurcations, chaotic behavior, sensitivity analysis and new optical solitons solutions of Sasa-Satsuma equation. Nonlinear Dyn. 2024;112(9):7405-7415. https://doi.org/10.1007/s11071-024-09438-6
  47. Demirbilek U, Muhammad J, Akbulut A, Younas U, Radwan T, Ahmed KK. The dynamical landscape of the negative-order (3+1)-dimensional Calogero-Bogoyavlenskii-Schiff equation. J Math. 2026;2026(1):8273647. https://doi.org/10.1155/jom/8273647
Share
Back to top
An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
We use cookies on our website to ensure you get the best experience.
Read more about our cookies here
Accept