Optical solitons with polynomial law of self-phase modulation by the fourth-order conservative finite difference scheme

Jiaqi Chen; Weizhong Dai; and Anjan Biswas; Jiaqi Chen; Weizhong Dai; and Anjan Biswas

doi:10.3934/math.2026474

AIMS Mathematics

2026, Volume 11, Issue 4: 11520-11545. doi: 10.3934/math.2026474

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Optical solitons with polynomial law of self-phase modulation by the fourth-order conservative finite difference scheme

1.
School of Electrical Engineering and Automation, Xiamen University of Technology, Xiamen, Fujian, 361024, China
2.
Mathematics & Statistics, College of Engineering & Science, Louisiana Tech University, Ruston, LA 71272, USA
3.
Department of Mathematics and Physics, Grambling State University, Grambling, LA 71245-2715, USA
4.
Department of Mathematics, Faculty of Science, Karadeniz Technical University, Trabzon, 61080, Türkiye
5.
Department of Physics and Electronics, Khazar University, Baku, AZ-1096, Azerbaijan
6.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa, Pretoria, 0204, South Africa

Received: 17 March 2026 Revised: 14 April 2026 Accepted: 21 April 2026 Published: 24 April 2026
MSC : 35Q55, 65M06, 78A60, 81V80

This paper considers an initial-boundary value problem for optical soliton propagation based on the cubic-quintic-septic nonlinear Schrödinger equation. We first analyzed the underlying physical properties of the system, formally establishing the conservation laws for mass and energy. Subsequently, we proposed a fourth-order finite difference scheme derived via the discrete variational derivative method to effectively approximate the nonlinear potential. We rigorously proved that the resulting numerical scheme perfectly satisfies the discrete analogues of the mass and energy conservation laws. Furthermore, we provided comprehensive stability and convergence analysis, demonstrating unconditional stability and establishing an error bound of $ O(\tau^2 + h^4) $. Numerical experiments were conducted to validate the theoretical analysis and illustrate the efficiency and high-order accuracy of the proposed approach.
- nonlinear Schrödinger equation,
- finite difference method,
- conservation law,
- stability,
- convergence
Citation: Jiaqi Chen, Weizhong Dai, and Anjan Biswas. Optical solitons with polynomial law of self-phase modulation by the fourth-order conservative finite difference scheme[J]. AIMS Mathematics, 2026, 11(4): 11520-11545. doi: 10.3934/math.2026474

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Abstract

This paper considers an initial-boundary value problem for optical soliton propagation based on the cubic-quintic-septic nonlinear Schrödinger equation. We first analyzed the underlying physical properties of the system, formally establishing the conservation laws for mass and energy. Subsequently, we proposed a fourth-order finite difference scheme derived via the discrete variational derivative method to effectively approximate the nonlinear potential. We rigorously proved that the resulting numerical scheme perfectly satisfies the discrete analogues of the mass and energy conservation laws. Furthermore, we provided comprehensive stability and convergence analysis, demonstrating unconditional stability and establishing an error bound of $ O(\tau^2 + h^4) $. Numerical experiments were conducted to validate the theoretical analysis and illustrate the efficiency and high-order accuracy of the proposed approach.

References

[1]	G. Agrawal, Fiber-optic communication systems, John Wiley & Sons, 2010. https://doi.org/10.1002/9780470918524
[2]	N. Akhmediev, A. Ankiewicz, Solitons: Nonlinear pulses and beams, New York: Springer, 1997.
[3]	T. Cazenave, Semilinear Schrödinger equations, American Mathematical Society, 2003.
[4]	A. Wazwaz, L. Kaur, Optical solitons for nonlinear Schrödinger (NLS) equation in normal dispersive regimes, Optik, 184 (2019), 428–435. https://doi.org/10.1016/j.ijleo.2019.04.118 doi: 10.1016/j.ijleo.2019.04.118
[5]	N. Kudryashov, Families of nonlinear Schrödinger equations in general form with exact solutions, Phys. Lett. A, 552 (2025), 130648. https://doi.org/10.1016/j.physleta.2025.130648 doi: 10.1016/j.physleta.2025.130648
[6]	N. Kudryashov, A. Polyanin, Nonlinear Schrödinger equations of general form and their exact solutions, Appl. Math. Lett., 170 (2025), 109622. https://doi.org/10.1016/j.aml.2025.109622 doi: 10.1016/j.aml.2025.109622
[7]	A. Biswas, Optical soliton cooling with polynomial law of nonlinear refractive index, J. Opt., 49 (2020), 580–583. https://doi.org/10.1007/s12596-020-00644-0 doi: 10.1007/s12596-020-00644-0
[8]	N. Akhmediev, A. Ankiewicz, R. Grimshaw, Hamiltonian-versus-energy diagrams in soliton theory, Phys. Rev. E, 59 (1999), 6088. https://doi.org/10.1103/physreve.59.6088 doi: 10.1103/physreve.59.6088
[9]	R. Shohib, M. Alngar, A. Arnous, A. Biswas, B. Rawal, Y. Yildirim, et al., Optical soliton parameters by variational principle: Polynomial and triple power-laws (super-Gaussian and super-sech pulses), Ukr. J. Phys. Opt., 25 (2024), 03068–03092. https://doi.org/10.3116/16091833/Ukr.J.Phys.Opt.2024.03068 doi: 10.3116/16091833/Ukr.J.Phys.Opt.2024.03068
[10]	D. Wang, Z. Liu, H. Zhao, H. Qin, G. Bai, C. Chen, et al., Launching by cavitation, Science, 389 (2025), 935–939. https://doi.org/10.1126/science.adu8943
[11]	Z. Si, D. Wang, B. Zhu, Z. Ju, X. Wang, W. Liu, et al., Deep learning for dynamic modeling and coded information storage of vector-soliton pulsations in mode-locked fiber lasers, Laser Photonics Rev., 18 (2024), 2400097. https://doi.org/10.1002/lpor.202400097 doi: 10.1002/lpor.202400097
[12]	Z. Si, Z. Ju, L. Ren, X. Wang, B. Malomed, C. Dai, Polarization-induced buildup and switching mechanisms for soliton molecules composed of noise-like-pulse transition states, Laser Photonics Rev., 19 (2025), 2401019. https://doi.org/10.1002/lpor.202401019 doi: 10.1002/lpor.202401019
[13]	I. Mendez Zuniga, T. Belyaeva, M. Agüero, V. Serkin, Multisoliton bound states in the fourth-order concatenation model of the nonlinear Schrödinger equation hierarchy, Trans. Opt. Photonics, 2026, 22–33.
[14]	D. Mou, Z. Si, W. Qiu, C. Dai, Optical soliton formation and dynamic characteristics in photonic Moiré lattices, Opt. Laser Technol., 181 (2025), 111774. https://doi.org/10.1016/j.optlastec.2024.111774 doi: 10.1016/j.optlastec.2024.111774
[15]	T. Belyaeva, L. Kovachev, V. Serkin, Parton-like soliton structures in nonlinear coherent states, Optik, 210 (2020), 164483. https://doi.org/10.1016/j.ijleo.2020.164483 doi: 10.1016/j.ijleo.2020.164483
[16]	J. Yang, Y. Zhu, W. Qin, S. Wang, C. Dai, J. Li, Higher-dimensional soliton structures of a variable-coefficient Gross-Pitaevskii equation with the partially nonlocal nonlinearity under a harmonic potential, Nonlinear Dyn., 108 (2022), 2551–2562. https://doi.org/10.1007/s11071-022-07337-2 doi: 10.1007/s11071-022-07337-2
[17]	J. Yang, Y. Zhu, W. Qin, S. Wang, J. Li, 3D bright-bright Peregrine triple-one structures in a nonautonomous partially nonlocal vector nonlinear Schrödinger model under a harmonic potential, Nonlinear Dyn., 111 (2023), 13287–13296. https://doi.org/10.1007/s11071-023-08526-3 doi: 10.1007/s11071-023-08526-3
[18]	K. Omar, F. Easif, Numerical solution of cubic-quintic nonlinear Schrödinger equation, Sci. J. Univ. Zakho, 13 (2025), 499–509. https://doi.org/10.25271/sjuoz.2025.13.4.1595 doi: 10.25271/sjuoz.2025.13.4.1595
[19]	H. Ibarra Villalon, O. Pottiez, A. Gómez Vieyra, J. Lauterio Cruz, Comparative study of finite difference methods and pseudo-spectral methods for solving the nonlinear Schrödinger equation in optical fiber, Phys. Scr., 98 (2023), 065514. https://doi.org/10.1088/1402-4896/acd22c doi: 10.1088/1402-4896/acd22c
[20]	Z. Z. Sun, Numerical solution methods for partial differential equations (in Chinese), Beijing: Science Press, 2005.
[21]	X. Wang, An energy-preserving finite difference scheme with fourth-order accuracy for the generalized Camassa-Holm equation, Commun. Nonlinear Sci. Numer. Simul., 119 (2023), 107–121. https://doi.org/10.1016/j.cnsns.2023.107121 doi: 10.1016/j.cnsns.2023.107121
[22]	W. Dai, R. Nassar, A finite difference scheme for the generalized nonlinear Schrödinger equation with variable coefficients, J. Comput. Math., 18 (2000), 123–132.
[23]	M. Delfour, M. Fortin, G. Payr, Finite-difference solutions of a non-linear Schrödinger equation, J. Comput. Phys., 44 (1981), 277–288. https://doi.org/10.1016/0021-9991(81)90052-8 doi: 10.1016/0021-9991(81)90052-8
[24]	X. Wang, W. Dai, A. Biswas, A conservative higher-order finite difference scheme for solving the Gardner equation with dual power-law nonlinearities in both 1D and 2D, Comput. Math. Appl., 201 (2026), 171–194. https://doi.org/10.1016/j.camwa.2025.10.023 doi: 10.1016/j.camwa.2025.10.023
[25]	L. Evans, Partial differential equations, American Mathematical Society, 2010.
[26]	X. Wang, H. Cheng, Solitary wave solution and a linear mass-conservative difference scheme for the generalized Korteweg-de Vries-Kawahara equation, Comput. Appl. Math., 40 (2021), 273. https://doi.org/10.1007/s40314-021-01668-3 doi: 10.1007/s40314-021-01668-3
[27]	N. Tamang, B. Wongsaijai, T. Mouktonglang, K. Poochinapan, Novel algorithm based on modification of Galerkin finite element method to general Rosenau-RLW equation in (2+1)-dimensions, Appl. Numer. Math., 148 (2020), 109–130. https://doi.org/10.1016/j.apnum.2019.07.021 doi: 10.1016/j.apnum.2019.07.021

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