Analytical Structure and Exact Solutions of a Generalized (2+1)-Dimensional Sasa–Satsuma Equation with Higher-Order Effects

İlker Burak Giresunlu

doi:10.36753/mathenot.1897916

Analytical Structure and Exact Solutions of a Generalized (2+1)-Dimensional Sasa–Satsuma Equation with Higher-Order Effects

Abstract

This study presents a unified analytical investigation of the generalized $(2+1)$-dimensional Sasa--Satsuma equation, which models nonlinear wave propagation in optical media. By employing a traveling wave transformation, the governing partial differential equation is reduced to a second-order ordinary differential equation. The intrinsic conservative structure of the reduced system is revealed through the derivation of an explicit first integral, which enables the construction of an effective potential function governing the wave dynamics. Based on this dynamical systems formulation, a fundamental bright soliton solution is obtained directly from the first integral, providing a structural interpretation of localized wave behavior. Furthermore, the $(G'/G,1/G)$-expansion method is applied to construct broader classes of exact analytical solutions, including hyperbolic, trigonometric, and rational forms. Unlike standard approaches where solution methods are applied independently, the obtained solutions are shown to be consistent with the intrinsic dynamical structure of the system. This establishes a direct connection between analytical solution techniques and dynamical systems theory, offering a more comprehensive understanding of nonlinear wave propagation in higher-dimensional Sasa--Satsuma-type models.

Keywords

Bifurcation structure, Effective potential, First integral, Kerr nonlinearity, Optical solitons, Traveling wave reduction

References

A. K. Sarma, Solitary wave solutions of higher-order NLSE with Raman and self-steepening effect in a cubic–quintic–septic medium, Commun. Nonlinear Sci. Numer. Simul., 14(8) (2009), 3215–3219. https://doi.org/10.1016/j.cnsns.2009.01.008
G. P. Agrawal, Nonlinear Fiber Optics, 5th ed., Academic Press, New York, 2012. https://doi.org/10.1007/3-540-46629-0_9
N. Sasa, J. Satsuma, New type of soliton solutions for a higher order nonlinear Schrödinger equation, J. Phys. Soc. Jpn., 60(2) (1991), 409–417. https://doi.org/10.1143/JPSJ.60.409
C. R. Menyuk, Pulse propagation in an elliptically birefringent Kerr medium, IEEE J. Quantum Electron., 25(12) (1989), 2674–2682. https://doi.org/10.1109/3.40656
C. Gilson, J. Hietarinta, J. Nimmo, et al., Sasa-Satsuma higher-order nonlinear Schrödinger equation and its bilinearization and multisoliton solutions, Phys. Rev. E, 68 (2003), Article ID 016614. https://doi.org/10.1103/PhysRevE.68.016614
C. Wu, B. Wei, C. Shi, et al., Multi-breather solutions to the Sasa–Satsuma equation, Proc. R. Soc. A, 478 (2022), Article ID 20210711. https://doi.org/10.1098/rspa.2021.0711
J. J. C. Nimmo, H. Yılmaz, Binary Darboux transformation for the Sasa–Satsuma equation, J. Phys. A: Math. Theor., 48(42) (2015), Article ID 425202. https://doi.org/10.1088/1751-8113/48/42/425202
Y. Ohta, J. Yang, General higher-order rogue waves and their dynamics in the Sasa-Satsuma equation, Proc. R. Soc. A, 468(2142) (2012), 1716–1740. https://doi.org/10.1098/rspa.2011.0640
A. Mvogo, L. F. Mouassom, F. A. Nyam, et al., Exact solitary waves for the 2D Sasa-Satsuma equation, Chaos Solitons Fractals, 135 (2020), Article ID 109657. https://doi.org/10.1016/j.chaos.2020.109657
Q. Yin, B. Gao, On complex soliton solutions, complex elliptic solutions and complex rational function solutions for the Sasa–Satsuma model equation with variable coefficients, J. Geom. Phys., 176 (2022), Article ID 104508. https://doi.org/10.1016/j.geomphys.2022.104508

M.Wang, X. Li, J. Zhang, The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372(4) (2008), 417–423. https://doi.org/10.1016/j.physleta.2007.07.051
A. Bekir, Application of the (G′/G)-expansion method for nonlinear evolution equations, Phys. Lett. A, 372(19) (2008), 3400–3406. https://doi.org/10.1016/j.physleta.2008.01.057
J. Chen, B. Li, Multiple (G′/G)-expansion method and its applications to nonlinear evolution equations in mathematical physics, Pramana, 78 (2012), 375–388. https://doi.org/10.1007/s12043-011-0237-6
S. Demiray, Ö. Ünsal, A. Bekir, Exact solutions of nonlinear wave equations using the (G′/G, 1/G)-expansion method, J. Egyptian Math. Soc., 23(1) (2015), 78–84. https://doi.org/10.1016/j.joems.2014.02.011
E. Yaşar, İ. B. Giresunlu, The (G′/G, 1/G)-expansion method for solving nonlinear space–time fractional differential equations, Pramana, 87 (2016), Article ID 17. https://doi.org/10.1007/s12043-016-1225-7
İ. B. Giresunlu, B. Yıldız, Analytical rational solitons of the modified Lakshmanan–Porsezian–Daniel equation, Univ. J. Math. Appl., 6(2) (2023), 53–64. https://doi.org/10.32323/ujma.1289530
A. R. Adem, Y. Yıldırım, E. Yaşar, Complexiton solutions and soliton solutions: (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation, Pramana, 92 (2019), Article ID 36. https://doi.org/10.1007/ s12043-018-1707-x
Y. Yıldırım, Sub pico-second pulses in mono-mode optical fibers with Triki-Biswas model using trial equation architecture, Optik, 183 (2019), 463–466. https://doi.org/10.1016/j.ijleo.2019.02.018
A. Biswas, Y. Yıldırım, E. Yaşar, et al., Optical soliton perturbation with complex Ginzburg–Landau equation using trial solution approach, Optik, 160 (2018), 44–60. https://doi.org/10.1016/j.ijleo.2018.01.102
Y. Yıldırım, E. Yaşar, Multiple exp-function method for soliton solutions of nonlinear evolution equations, Chinese Physics B, 26(7) (2017), Article ID 070201. https://doi.org/10.1088/1674-1056/26/7/070201
A. H. Arnous, A. Biswas, A. H. Kara, et al., Optical solitons and conservation laws for the concatenation model: Power–law nonlinearity, Ain Shams Eng. J., 15(2) (2024), Article ID 102381. https://doi.org/10.1016/j.asej.2023.102381
N. Özdemir, A. Seçer, M. Özışık, et al., Optical soliton solutions of the nonlinear Schrödinger equation in the presence of chromatic dispersion with cubic–quintic–septic–nonic nonlinearities, Phys. Scripta, 98 (2023), Article ID 115223. https://doi.org/10.1088/1402-4896/acff50
İ. Önder, H. Esen, A. Seçer, et al., Stochastic optical solitons of the perturbed nonlinear Schrödinger equation with Kerr law via Ito calculus, Eur. Phys. J. Plus, 138 (2023), Article ID 872. https://doi.org/10.1140/epjp/s13360-023-04497-x
A. H. Arnous, A. M. Elsherbeny, A. Seçer, et al., Optical solitons for the dispersive concatenation model with spatio-temporal dispersion having multiplicative white noise, Results Phys., 56 (2024), Article ID 107299. https://doi.org/10.1016/j.rinp.2023.107299
C. Wu, G. Zhang, C. Shi, et al., General rogue wave solutions to the Sasa–Satsuma equation, IMA J. Appl. Math., 89(6) (2025), 953–975. https://doi.org/10.1093/imamat/hxaf003
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
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.
M. Çınar, A. Seçer, M. Özışık, et al., Optical soliton solutions of (1+1)- and (2+1)-dimensional generalized Sasa–Satsuma equations using new Kudryashov method, Int. J. Geom. Methods Mod. Phys., 20(2) (2023), Article ID 2350034. https://doi.org/10.1142/S0219887823500342
E. M. Zayed, M. E. Alngar, R. M. Shohib, et al., Optical solitons in birefringent fibers with Sasa–Satsuma equation having multiplicative noise with Itô calculus, J. Nonlinear Opt. Phys. Mater., 32(1) (2023), Article ID 2350006. https://doi.org/10.1142/S0218863523500066

Details

Primary Language

English

Subjects

Mathematical Methods and Special Functions

Journal Section

Research Article

Authors

İlker Burak Giresunlu ^*
0000-0002-2190-0003
Türkiye

Early Pub Date

May 9, 2026

Publication Date

June 3, 2026

Submission Date

February 25, 2026

Acceptance Date

May 5, 2026

Published in Issue

Year 2026 Volume: 14 Number: 2

DOI

https://doi.org/10.36753/mathenot.1897916

IZ

https://izlik.org/JA33WA83WR

APA

Giresunlu, İ. B. (2026). Analytical Structure and Exact Solutions of a Generalized (2+1)-Dimensional Sasa–Satsuma Equation with Higher-Order Effects. Mathematical Sciences and Applications E-Notes, 14(2), 106-123. https://doi.org/10.36753/mathenot.1897916

AMA

1.Giresunlu İB. Analytical Structure and Exact Solutions of a Generalized (2+1)-Dimensional Sasa–Satsuma Equation with Higher-Order Effects. Math. Sci. Appl. E-Notes. 2026;14(2):106-123. doi:10.36753/mathenot.1897916

Chicago

Giresunlu, İlker Burak. 2026. “Analytical Structure and Exact Solutions of a Generalized (2+1)-Dimensional Sasa–Satsuma Equation With Higher-Order Effects”. Mathematical Sciences and Applications E-Notes 14 (2): 106-23. https://doi.org/10.36753/mathenot.1897916.

EndNote

Giresunlu İB (June 1, 2026) Analytical Structure and Exact Solutions of a Generalized (2+1)-Dimensional Sasa–Satsuma Equation with Higher-Order Effects. Mathematical Sciences and Applications E-Notes 14 2 106–123.

IEEE

[1]İ. B. Giresunlu, “Analytical Structure and Exact Solutions of a Generalized (2+1)-Dimensional Sasa–Satsuma Equation with Higher-Order Effects”, Math. Sci. Appl. E-Notes, vol. 14, no. 2, pp. 106–123, June 2026, doi: 10.36753/mathenot.1897916.

ISNAD

Giresunlu, İlker Burak. “Analytical Structure and Exact Solutions of a Generalized (2+1)-Dimensional Sasa–Satsuma Equation With Higher-Order Effects”. Mathematical Sciences and Applications E-Notes 14/2 (June 1, 2026): 106-123. https://doi.org/10.36753/mathenot.1897916.

JAMA

1.Giresunlu İB. Analytical Structure and Exact Solutions of a Generalized (2+1)-Dimensional Sasa–Satsuma Equation with Higher-Order Effects. Math. Sci. Appl. E-Notes. 2026;14:106–123.

MLA

Giresunlu, İlker Burak. “Analytical Structure and Exact Solutions of a Generalized (2+1)-Dimensional Sasa–Satsuma Equation With Higher-Order Effects”. Mathematical Sciences and Applications E-Notes, vol. 14, no. 2, June 2026, pp. 106-23, doi:10.36753/mathenot.1897916.

Vancouver

1.İlker Burak Giresunlu. Analytical Structure and Exact Solutions of a Generalized (2+1)-Dimensional Sasa–Satsuma Equation with Higher-Order Effects. Math. Sci. Appl. E-Notes. 2026 Jun. 1;14(2):106-23. doi:10.36753/mathenot.1897916