This work investigates the generalized nonlinear Schrödinger equation (NLSE), which imitates the wave transmission along optical fibers. This model incorporates a quintuple power-law of non-linearity and nonlinear chromatic dispersion. To demonstrate the significance and motivation for this work, a review of the prior research is presented in the literature. Three integration strategies are applied during the study process in order to produce a variety of novel solutions. These techniques include the modified exp-function approach, the general projective Riccati method (GPRM), and the new generalized method. The extracted solutions include bright solitons, singular solitons, dark solitons, and trigonometric solutions.
Citation: Islam Samir, Hamdy M. Ahmed, Wafaa Rabie, W. Abbas, Ola Mostafa. Construction optical solitons of generalized nonlinear Schrödinger equation with quintuple power-law nonlinearity using Exp-function, projective Riccati, and new generalized methods[J]. AIMS Mathematics, 2025, 10(2): 3392-3407. doi: 10.3934/math.2025157
This work investigates the generalized nonlinear Schrödinger equation (NLSE), which imitates the wave transmission along optical fibers. This model incorporates a quintuple power-law of non-linearity and nonlinear chromatic dispersion. To demonstrate the significance and motivation for this work, a review of the prior research is presented in the literature. Three integration strategies are applied during the study process in order to produce a variety of novel solutions. These techniques include the modified exp-function approach, the general projective Riccati method (GPRM), and the new generalized method. The extracted solutions include bright solitons, singular solitons, dark solitons, and trigonometric solutions.
[1] |
Y. Sun, Z. Hu, H. Triki, M. Mirzazadeh, W. Liu, A. Biswas, et al., Analytical study of three-soliton interactions with different phases in nonlinear optics, Nonlinear Dyn., 111 (2023), 18391–18400. https://doi.org/10.1007/s11071-023-08786-z doi: 10.1007/s11071-023-08786-z
![]() |
[2] |
Y. Zhong, K. Yu, Y. Sun, H. Triki, Q. Zhou, Stability of solitons in Bose–Einstein condensates with cubic–quintic–septic nonlinearity and non-PT-symmetric complex potentials, Eur. Phys. J. Plus, 139 (2024), 119. https://doi.org/10.1140/epjp/s13360-024-04930-9 doi: 10.1140/epjp/s13360-024-04930-9
![]() |
[3] |
Q. Li, W. Zou, Normalized ground states for Sobolev critical nonlinear Schrödinger equation in the $L^2$-supercritical case, DCDS, 44 (2024), 205–227. https://doi.org/10.3934/dcds.2023101 doi: 10.3934/dcds.2023101
![]() |
[4] |
S. Malik, S. Kumar, Pure-cubic optical soliton perturbation with full nonlinearity by a new generalized approach, Optik, 258 (2022), 168865. https://doi.org/10.1016/j.ijleo.2022.168865 doi: 10.1016/j.ijleo.2022.168865
![]() |
[5] |
Q. Li, J. Nie, W. Wang, J. Zhou, Normalized solutions for Sobolev critical fractional Schrödinger equation, Adv. Nonlinear Anal., 13 (2024), 20240027. https://doi.org/10.1515/anona-2024-0027 doi: 10.1515/anona-2024-0027
![]() |
[6] | W. Zhang, J. Zhang, V. Rădulescu, Semiclassical states for the pseudo-relativistic Schrödinger equation with competing potentials, Commun. Math. Sci., 23 (2025), 465–507. |
[7] |
S. Shen, Z. J. Yang, Z. G. Pang, Y. R. Ge, The complex-valued astigmatic cosine-Gaussian soliton solution of the nonlocal nonlinear Schrödinger equation and its transmission characteristics, Appl. Math. Lett., 125 (2022), 107755. https://doi.org/10.1016/j.aml.2021.107755 doi: 10.1016/j.aml.2021.107755
![]() |
[8] |
M. A. E. Abdelrahman, A. Alharbi, M. B. Almatrafi, Fundamental solutions for the generalised third-order nonlinear Schrödinger equation, Int. J. Appl. Comput. Math., 6 (2020), 160. https://doi.org/10.1007/s40819-020-00906-2 doi: 10.1007/s40819-020-00906-2
![]() |
[9] |
W. B. Rabie, H. M. Ahmed, Construction cubic-quartic solitons in optical metamaterials for the perturbed twin-core couplers with Kudryashov's sextic power law using extended F-expansion method, Chaos Soliton Fract., 160 (2022), 112289. https://doi.org/10.1016/j.chaos.2022.112289 doi: 10.1016/j.chaos.2022.112289
![]() |
[10] |
I. Onder, A. Secer, M. Ozisik, M. Bayram, Investigation of optical soliton solutions for the perturbed Gerdjikov-Ivanov equation with full-nonlinearity, Heliyon, 9 (2023), e13519. https://doi.org/10.1016/j.heliyon.2023.e13519 doi: 10.1016/j.heliyon.2023.e13519
![]() |
[11] |
I. Samir, H. M. Ahmed, S. Alkhatib, E. M. Mohamed, Construction of wave solutions for stochastic Radhakrishnan–Kundu–Lakshmanan equation using modified extended direct algebraic technique, Results Phys., 55 (2023), 107191. https://doi.org/10.1016/j.rinp.2023.107191 doi: 10.1016/j.rinp.2023.107191
![]() |
[12] | R. Kumar, R. Kumar, A. Bansal, A. Biswas, Y. Yildirim, S. P. Moshokoa, et al., Optical solitons and group invariants for Chen-Lee-Liu equation with time-dependent chromatic dispersion and nonlinearity by Lie symmetry, Ukr. J. Phys. Opt., 2023. |
[13] |
M. M. Khatun, M. A. Akbar, New optical soliton solutions to the space-time fractional perturbed Chen-Lee-Liu equation, Results Phys., 46 (2023), 106306. https://doi.org/10.1016/j.rinp.2023.106306 doi: 10.1016/j.rinp.2023.106306
![]() |
[14] |
M. Sadaf, G. Akram, S. Arshed, K. Farooq, A study of fractional complex Ginzburg–Landau model with three kinds of fractional operators, Chaos Soliton. Fract., 166 (2023), 112976. https://doi.org/10.1016/j.chaos.2022.112976 doi: 10.1016/j.chaos.2022.112976
![]() |
[15] |
K. J. Wang, J. Si, Diverse optical solitons to the complex Ginzburg–Landau equation with Kerr law nonlinearity in the nonlinear optical fiber, Eur. Phys. J. Plus, 138 (2023), 187. https://doi.org/10.1140/epjp/s13360-023-03804-w doi: 10.1140/epjp/s13360-023-03804-w
![]() |
[16] |
W. Chen, J. Manafian, K. H. Mahmoud, A. S. Alsubaie, A. Aldurayhim, A. Alkader, Cutting-edge analytical and numerical approaches to the Gilson–Pickering equation with plenty of soliton solutions, Mathematics, 11 (2023), 3454. https://doi.org/10.3390/math11163454 doi: 10.3390/math11163454
![]() |
[17] |
A. Yokuş, H. Durur, K. A. Abro, D. Kaya, Role of Gilson–Pickering equation for the different types of soliton solutions: a nonlinear analysis, Eur. Phys. J. Plus, 135 (2020), 657. https://doi.org/10.1140/epjp/s13360-020-00646-8 doi: 10.1140/epjp/s13360-020-00646-8
![]() |
[18] |
N. A. Kudryashov, The Lakshmanan–Porsezian–Daniel model with arbitrary refractive index and its solution, Optik, 241 (2021), 167043. https://doi.org/10.1016/j.ijleo.2021.167043 doi: 10.1016/j.ijleo.2021.167043
![]() |
[19] |
G. Liang, H. Zhang, L. Fang, Q. Shou, W. Hu, Q. Guo, Influence of transverse cross-phases on propagations of optical beams in linear and nonlinear regimes, Laser Photonics Rev., 14 (2020), 2000141. https://doi.org/10.1002/lpor.202000141 doi: 10.1002/lpor.202000141
![]() |
[20] |
I. Samir, H. M. Ahmed, A. Darwish, H. H. Hussein, Dynamical behaviors of solitons for NLSE with Kudryashov's sextic power-law of nonlinear refractive index using improved modified extended tanh-function method, Ain Shams Eng. J., 15 (2024), 102267. https://doi.org/10.1016/j.asej.2023.102267 doi: 10.1016/j.asej.2023.102267
![]() |
[21] |
O. El-shamy, R. El-barkoki, H. M. Ahmed, W. Abbas, I. Samir, Exploration of new solitons in optical medium with higher-order dispersive and nonlinear effects via improved modified extended tanh function method, Alex. Eng. J., 68 (2023), 611–618. https://doi.org/10.1016/j.aej.2023.01.053 doi: 10.1016/j.aej.2023.01.053
![]() |
[22] |
I. Samir, A. Abd-Elmonem, H. M. Ahmed, General solitons for eighth-order dispersive nonlinear Schrödinger equation with ninth-power law nonlinearity using improved modified extended tanh method, Opt. Quant. Electron., 55 (2023), 470. https://doi.org/10.1007/s11082-023-04753-5 doi: 10.1007/s11082-023-04753-5
![]() |
[23] |
Y. Shang, The extended hyperbolic function method and exact solutions of the long–short wave resonance equations, Chaos Soliton. Fract., 36 (2008), 762–771. https://doi.org/10.1016/j.chaos.2006.07.007 doi: 10.1016/j.chaos.2006.07.007
![]() |
[24] |
L. Akinyemi, Two improved techniques for the perturbed nonlinear Biswas–Milovic equation and its optical solitons, Optik, 243 (2021), 167477. https://doi.org/10.1016/j.ijleo.2021.167477 doi: 10.1016/j.ijleo.2021.167477
![]() |
[25] |
N. Savaissou, B. Gambo, H. Rezazadeh, A. Bekir, S. Y. Doka, Exact optical solitons to the perturbed nonlinear Schrödinger equation with dual-power law of nonlinearity, Opt. Quant. Electron., 52 (2020), 318. https://doi.org/10.1007/s11082-020-02412-7 doi: 10.1007/s11082-020-02412-7
![]() |
[26] |
E. A. Az-Zo'bi, W. A. Alzoubi, L. Akinyemi, M. Şenol, B. S. Masaedeh, A variety of wave amplitudes for the conformable fractional (2+1)-dimensional Ito equation, Mod. Phys. Lett. B, 35 (2021), 2150254. https://doi.org/10.1142/S0217984921502547 doi: 10.1142/S0217984921502547
![]() |
[27] |
W. B. Rabie, H. M. Ahmed, I. Samir, M. Alnahhass, Optical solitons and stability analysis for NLSE with nonlocal nonlinearity, nonlinear chromatic dispersion and Kudryashov's generalized quintuple-power nonlinearity, Results Phys., 59 (2024), 107589. https://doi.org/10.1016/j.rinp.2024.107589 doi: 10.1016/j.rinp.2024.107589
![]() |
[28] |
A. Biswas, M. Ekici, A. Sonmezoglu, Stationary optical solitons with Kudryashov's quintuple power–law of refractive index having nonlinear chromatic dispersion, Phys. Lett. A, 426 (2022), 127885. https://doi.org/10.1016/j.physleta.2021.127885 doi: 10.1016/j.physleta.2021.127885
![]() |
[29] |
M. Ekici, Stationary optical solitons with complex Ginzburg–Landau equation having nonlinear chromatic dispersion and Kudryashov's refractive index structures, Phys. Lett. A, 440 (2022), 128146. https://doi.org/10.1016/j.physleta.2022.128146 doi: 10.1016/j.physleta.2022.128146
![]() |
[30] |
A. Sonmezoglu, Stationary optical solitons having Kudryashov's quintuple power law nonlinearity by extended $G'/G$–expansion, Optik, 253 (2022), 168521. https://doi.org/10.1016/j.ijleo.2021.168521 doi: 10.1016/j.ijleo.2021.168521
![]() |
[31] |
M. G. Hafez, M. A. Akbar, An exponential expansion method and its application to the strain wave equation in microstructured solids, Ain Shams Eng. J., 6 (2015), 683–690. https://doi.org/10.1016/j.asej.2014.11.011 doi: 10.1016/j.asej.2014.11.011
![]() |
[32] |
G. Akram, M. Sadaf, S. Arshed, F. Sameen, Bright, dark, kink, singular and periodic soliton solutions of Lakshmanan–Porsezian–Daniel model by generalized projective Riccati equations method, Optik, 241 (2021), 167051. https://doi.org/10.1016/j.ijleo.2021.167051 doi: 10.1016/j.ijleo.2021.167051
![]() |