Analytical solutions, bifurcations, and chaotic patterns of the Sasa-Satsuma equation via the $ exp(-\Phi(\zeta)) $ expansion method

B. Alreshidi; Abdulaziz Almaslokh; A.A. Elsadany; Mohammed. K. Elboree; B. Alreshidi; Abdulaziz Almaslokh; A.A. Elsadany; Mohammed. K. Elboree

doi:10.3934/math.2026632

AIMS Mathematics

2026, Volume 11, Issue 6: 15376-15401. doi: 10.3934/math.2026632

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Analytical solutions, bifurcations, and chaotic patterns of the Sasa-Satsuma equation via the $ exp(-\Phi(\zeta)) $ expansion method

1.
Department of Mathematics, Faculty of Sciences and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Qena University, Qena 83523, Egypt

Received: 30 March 2026 Revised: 06 May 2026 Accepted: 21 May 2026 Published: 01 June 2026
MSC : 35Q51, 35Q53, 37K40

In this study, the propagation of ultrashort optical pulses in nonlinear dispersive media with third-order dispersion, self-steepening, and stimulated Raman scattering is modelled using the Sasa-Satsuma higher-order nonlinear Schrödinger equation. Exact travelling wave solutions, including bright solitons, dark solitons, singular solutions, and periodic wave solutions, are obtained by applying a travelling wave transformation and applying the $ \exp(-\Phi(\zeta)) $ function method on the resulting ordinary differential equation. A systematic bifurcation analysis of the associated planar Hamiltonian dynamical system validates these solutions, identifying equilibrium points, constructing phase portraits across all parameter regimes, and explicitly linking orbit families to solution types. Chaotic dynamics are analyzed using time series and phase diagrams. Using Poincaré sections, Lyapunov exponent spectra, time series, and bifurcation diagrams, we map the transition from regular to chaotic regimes and demonstrate how sensitivity to initial conditions governs predictability. The sensitivity analysis shows us the point at which the system becomes very sensitive to changes in the parameter values, which measures how uncertainties in initial conditions and parameters propagate and grow over time in the system.
- Sasa-Satsuma equation,
- higher-order nonlinear Schrödinger equation,
- bifurcation analysis,
- phase portrait,
- chaotic dynamics,
- traveling wave solutions,
- Poincaré section
Citation: B. Alreshidi, Abdulaziz Almaslokh, A.A. Elsadany, Mohammed. K. Elboree. Analytical solutions, bifurcations, and chaotic patterns of the Sasa-Satsuma equation via the $ exp(-\Phi(\zeta)) $ expansion method[J]. AIMS Mathematics, 2026, 11(6): 15376-15401. doi: 10.3934/math.2026632

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Abstract

In this study, the propagation of ultrashort optical pulses in nonlinear dispersive media with third-order dispersion, self-steepening, and stimulated Raman scattering is modelled using the Sasa-Satsuma higher-order nonlinear Schrödinger equation. Exact travelling wave solutions, including bright solitons, dark solitons, singular solutions, and periodic wave solutions, are obtained by applying a travelling wave transformation and applying the $ \exp(-\Phi(\zeta)) $ function method on the resulting ordinary differential equation. A systematic bifurcation analysis of the associated planar Hamiltonian dynamical system validates these solutions, identifying equilibrium points, constructing phase portraits across all parameter regimes, and explicitly linking orbit families to solution types. Chaotic dynamics are analyzed using time series and phase diagrams. Using Poincaré sections, Lyapunov exponent spectra, time series, and bifurcation diagrams, we map the transition from regular to chaotic regimes and demonstrate how sensitivity to initial conditions governs predictability. The sensitivity analysis shows us the point at which the system becomes very sensitive to changes in the parameter values, which measures how uncertainties in initial conditions and parameters propagate and grow over time in the system.

References

[1]	V. E. Zakharov, A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP, 34 (1972), 62–69.
[2]	A. Hasegawa, F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. Ⅰ. Anomalous dispersion, Appl. Phys. Lett., 23 (1973), 142–144. https://doi.org/10.1063/1.1654836 doi: 10.1063/1.1654836
[3]	M. J. Ablowitz, H. Segur, Solitons and the inverse scattering transform, Philadelphia: SIAM, 1981. https://doi.org/10.1137/1.9781611970883
[4]	G. P. Agrawal, Nonlinear fiber optics, London; San Diego, CA: Academic Press, 2019. https://doi.org/10.1016/C2018-0-01168-8
[5]	Y. Kodama, A. Hasegawa, Nonlinear pulse propagation in a monomode dielectric guide, IEEE J. Quantum Electron., 23 (1987), 510–524. https://doi.org/10.1109/JQE.1987.1073392 doi: 10.1109/JQE.1987.1073392
[6]	N. Sasa, J. Satsuma, New-type of soliton solutions for a higher-order nonlinear Schrödinger equation, J. Phys. Soc. Jpn., 60 (1991), 409–417. https://doi.org/10.1143/JPSJ.60.409 doi: 10.1143/JPSJ.60.409
[7]	Y. S. Kivshar, G. P. Agrawal, Optical solitons: from fibers to photonic crystals, San Diego, CA: Academic Press, 2003. https://doi.org/10.1016/B978-0-12-410590-4.X5000-1
[8]	J. J. C. Nimmo, H. Yilmaz, Binary darboux transformation for the Sasa-Satsuma equation, J. Phys. A: Math. Theor., 48 (2015), 425202. https://doi.org/10.1088/1751-8113/48/42/425202 doi: 10.1088/1751-8113/48/42/425202
[9]	J. Xu, E. Fan, The unified transform method for the Sasa-Satsuma equation on the half-line, Proc. R. Soc. A, 469 (2013), 20130068. https://doi.org/10.1098/rspa.2013.0068 doi: 10.1098/rspa.2013.0068
[10]	J. M. Soto-Crespo, N. Devine, N. P. Hoffmann, N. Akhmediev, Rogue waves of the Sasa-Satsuma equation in a chaotic wave field, Phys. Rev. E, 90 (2014), 032902. https://doi.org/10.1103/PhysRevE.90.032902 doi: 10.1103/PhysRevE.90.032902
[11]	S. Chen, F. Baronio, J. M. Soto-Crespo, P. Grelu, D. Mihalache, Versatile rogue waves in scalar, vector, and multidimensional nonlinear systems, J. Phys. A: Math. Theor., 50 (2017), 463001. https://doi.org/10.1088/1751-8121/aa8f00 doi: 10.1088/1751-8121/aa8f00
[12]	J. Li, Singular nonlinear traveling wave equations: bifurcations and exact solutions, Beijing: Science Press, 2013.
[13]	J. Li, G. Chen, On a class of singular nonlinear traveling wave equations, Int. J. Bifurc. Chaos, 17 (2007), 4049–4065. https://doi.org/10.1142/S0218127407019858 doi: 10.1142/S0218127407019858
[14]	J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, New York: Springer, 1983. https://doi.org/10.1007/978-1-4612-1140-2
[15]	L. Tang, Bifurcation analysis and multiple solitons in birefringent fibers with coupled Schrödinger–Hirota equation, Chaos Soliton. Fract., 161 (2022), 112383. https://doi.org/10.1016/j.chaos.2022.112383 doi: 10.1016/j.chaos.2022.112383
[16]	S. H. Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Boulder: Westview Press, 2015.
[17]	G. C. Layek, An introduction to dynamical systems and chaos, New Delhi: Springer, 2015. https://doi.org/10.1007/978-81-322-2556-0
[18]	Z. Li, C. Huang, Bifurcation, phase portrait, chaotic pattern and optical soliton solutions of the conformable Fokas–Lenells model in optical fibers, Chaos Soliton. Fract., 169 (2023), 113237. https://doi.org/10.1016/j.chaos.2023.113237 doi: 10.1016/j.chaos.2023.113237
[19]	L. Tang, Phase portraits and multiple optical solitons perturbation in optical fibers with the nonlinear Fokas–Lenells equation, J. Opt., 52 (2023), 2214–2223. https://doi.org/10.1007/s12596-023-01097-x doi: 10.1007/s12596-023-01097-x
[20]	Z. Li, C. Huang, B. J. Wang, Phase portrait, bifurcation, chaotic pattern and optical soliton solutions of the Fokas–Lenells equation with cubic-quartic dispersion in optical fibers, Phys. Lett. A, 465 (2023), 128714. https://doi.org/10.1016/j.physleta.2023.128714 doi: 10.1016/j.physleta.2023.128714
[21]	C. Sulem, P. L. Sulem, The nonlinear Schrödinger equation: self-focusing and wave collapse, New York: Springer, 1999. https://doi.org/10.1007/b98958
[22]	C. Gilson, J. Hietarinta, J. Nimmo, Y. Ohta, Sasa-Satsuma higher-order nonlinear Schrödinger equation and its bilinearization and multisoliton solutions, Phys. Rev. E, 68 (2003), 016614. https://doi.org/10.1103/PhysRevE.68.016614 doi: 10.1103/PhysRevE.68.016614
[23]	X. Geng, J. P. Wu, Riemann–Hilbert approach and N-soliton solutions for a generalized Sasa-Satsuma equation, Wave Motion, 60 (2016), 62–72. https://doi.org/10.1016/j.wavemoti.2015.09.003 doi: 10.1016/j.wavemoti.2015.09.003
[24]	M. Shakeel, Attaullah, N. A. Shah, J. D. Chung, Modified exp-function method to find exact solutions of microtubules nonlinear dynamics models, Symmetry, 15 (2023), 360. https://doi.org/10.3390/sym15020360 doi: 10.3390/sym15020360
[25]	M. Shakeel, Attaullah, E. R. El-Zahar, N. A. Shah, J. D. Chung, Generalized exp-function method to find closed form solutions of nonlinear dispersive modified Benjamin–Bona–Mahony equation defined by seismic sea waves, Mathematics, 10 (2022), 1026. https://doi.org/10.3390/math10071026 doi: 10.3390/math10071026
[26]	M. I. Khan, A. Farooq, K. S. Nisar, N. A. Shah, Unveiling new exact solutions of the unstable nonlinear Schrödinger equation using the improved modified Sardar sub-equation method, Results Phys., 59 (2024), 107593. https://doi.org/10.1016/j.rinp.2024.107593 doi: 10.1016/j.rinp.2024.107593

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