Abstract
In this article, we aim to investigate the nonlinear Schrödinger equation that describes the pulse propagation in optical fiber through two novel techniques, namely, the Bäcklund transformation-based method and Wang’s direct mapping method for the first time. Diverse soliton solutions expressed in the form of trigonometric function such as sine, cosine, secant, cosecant, hyperbolic function like hyperbolic tangent, hyperbolic secant, hyperbolic cosecant, hyperbolic sine, hyperbolic cosine, exponential function and the rational function are obtained. The performances of the different soliton solutions are illustrated through the 3-D plots, 2-D contours and 2-D curves. It is confirmed that the proposed methods are powerful and effective, which can be used to study the other PDEs arising in optics.
Similar content being viewed by others
Data Availability Statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
M. Sohail, R. Naz, Z. Shah et al., Exploration of temperature dependent thermophysical characteristics of yield exhibiting non-Newtonian fluid flow under gyrotactic microorganisms. AIP Adv. 9(12), 125016 (2019)
K.J. Wang, J. Si, On the non-differentiable exact solutions of the (2+1)-dimensional local fractional breaking soliton equation on Cantor sets. Math. Methods Appl. Sci. 46(2), 1456–1465 (2023)
A. Biswas, D. Milovic, R. Koh, Optical soliton perturbation in a log-law medium with full nonlinearity by He’s semi-inverse variational principle. Inverse. Probl. Sci. Eng. 20(2), 227–232 (2021)
K.J. Wang, J. Si, Dynamic properties of the attachment oscillator arising in the nanophysics. Open Phys. (2023). https://doi.org/10.1515/phys-2022-0214
C.H. He, D. Tian, G.M. Moatimid, H.F. Salman, M.H. Zekry, Hybrid Rayleigh-Van der Pol-Duffing oscillator (HRVD): stability analysis and controller. J. Low Freq. Noise Vib. Act. Control 41(1), 244–268 (2022)
K.J. Wang, The fractal active Low-pass filter within the local fractional derivative on the Cantor set, COMPEL. Int. J. Comput. Math. Electr. Electron. Eng. (2023). https://doi.org/10.1108/COMPEL-09-2022-0326
K.J. Wang, On a High-pass filter described by local fractional derivative. Fractals 28(3), 2050031 (2020)
X. Lü, H.W. Hui, F.F. Liu, Y.L. Bai, Stability and optimal control strategies for a novel epidemic model of COVID-19. Nonlinear Dyn. 106, 1491 (2021)
D. Kumar, J. Singh, M.A. Qurashi, D. Baleanu, A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying. Adv. Differ. Equ 2019, 1–9 (2019)
A.H. Bhrawy, M.S. Alhuthali, M.A. Abdelkawy, New solutions for (1+ 1)-dimensional and (2+1)-dimensional Ito equations. Math. Probl. Eng. (2012). https://doi.org/10.1155/2012/537930
K.J. Wang, F. Shi, J.H. Liu, J. Si, Application of the extended F-expansion method for solving the fractional Gardner equation with conformable fractional derivative. Fractal 30(7), 2250139 (2022)
M.A. Abdou, The extended F-expansion method and its application for a class of nonlinear evolution equations. Chaos Solitons Fractals 31(1), 95–104 (2007)
W.B. Rabie, H.M. Ahmed, Cubic-quartic optical solitons and other solutions for twin-core couplers with polynomial law of nonlinearity using the extended F-expansion method. Optik 253, 168575 (2022)
D. Shang, Exact solutions of coupled nonlinear Klein-Gordon equation. Appl. Math. Comput. 217(4), 1577–1583 (2010)
E.M.E. Zayed, K.A. Gepreel, M. El-Horbaty et al., Optical solitons in birefringent fibers with Kaup-Newell equation using two integration schemes. Optik 251, 167992 (2022)
M. Cinar, I. Onder, A. Secer et al., The analytical solutions of Zoomeron equation via extended rational sin-cos and sinh-cosh methods. Phys. Scr. 96(9), 094002 (2021)
K.J. Wang, J.H. Liu, J. Wu, Soliton solutions to the Fokas system arising in monomode optical fibers. Optik 251, 168319 (2022)
N. Raza, A. Javid, Optical dark and dark-singular soliton solutions of (1+ 2)-dimensional chiral nonlinear Schrodinger’s equation. Waves Random Complex Media 29(3), 496–508 (2019)
S.T. Mohyud-Din, Y. Khan, N. Faraz et al., Exp-function method for solitary and periodic solutions of Fitzhugh-Nagumo equation. Int. J. Numer. Meth. Heat Fluid Flow 22(3), 335–341 (2012)
J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30(3), 700–708 (2006)
K.J. Wang, H.W. Zhu, Periodic wave solution of the Kundu-Mukherjee-Naskar equation in birefringent fibers via the Hamiltonian-based algorithm. EPL 139(3), 35002 (2022)
A.R. Seadawy, D. Kumar, A.K. Chakrabarty, Dispersive optical soliton solutions for the hyperbolic and cubic-quintic nonlinear Schrödinger equations via the extended sinh-Gordon equation expansion method. Eur. Phys. J. Plus 133(5), 182 (2018)
N. Raza, S. Arshed, S. Sial, Optical solitons for coupled Fokas-Lenells equation in birefringence fibers. Mod. Phys. Lett. B 33(26), 1950317 (2019)
K.J. Wang, Diverse soliton solutions to the Fokas system via the Cole-Hopf transformation. Optik 272, 170250 (2023)
K.J. Wang, J.H. Liu, Diverse optical soliton solutions to the Kundu-Mukherjee-Naskar equation via two novel techniques. Optik 273, 170403 (2023)
U. Afzal, N. Raza, I.G. Murtaza, On soliton solutions of time fractional form of Sawada-Kotera equation. Nonlinear Dyn. 95(1), 391–405 (2019)
K.J. Wang, J. Si, Optical solitons to the Radhakrishnan-Kundu-Lakshmanan equation by two effective approaches. Eur. Phys. J. Plus 137, 1016 (2022)
H. Rezazadeh, M. Inc, D. Baleanu, New solitary wave solutions for variants of (3+1)-dimensional Wazwaz-Benjamin-Bona-Mahony equations. Front. Phys. 8, 332 (2020)
K.J. Wang, J.H. Liu, On abundant wave structures of the unsteady korteweg-devries equation arising in shallow water. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.04.024
K.-J. Wang, J.-H. Liu, J. Si, G.-D. Wang, Nonlinear dynamic behaviors of the (3+1)-dimensional B-type Kadomtsev-Petviashvili equation in fluid mechanics. Axioms 12(1), 95 (2023)
Y. SaglamOzkan, A.R. Seadawy, E. Yasar, Multi-wave breather and interaction solutions to (3+1) dimensional Vakhnenko-Parkes equation arising at propagation of high-frequency waves in a relaxing medium. J. Taibah Univ. Sci. 15(1), 666–678 (2021)
A.I. Aliyu, M. Inc, A. Yusuf et al., Dark-bright optical soliton and conserved vectors to the Biswas-Arshed equation with third-order dispersions in the absence of self-phase modulation. Front. Phys. 7, 28 (2019)
N. Ozdemir, H. Esen, A. Secer et al., Optical Soliton Solutions to Chen Lee Liu model by the modified extended tanh expansion scheme. Optik 245, 167643 (2021)
K.J. Wang, F. Shi, G.D. Wang, Periodic wave structure of the fractal generalized fourth order Boussinesq equation travelling along the non-smooth boundary. Fractals 30(9), 2250168 (2022)
K.J. Wang, F. Shi, A new perspective on the exact solutions of the local fractional modified Benjamin-Bona-Mahony equation on Cantor sets. Fractal Fractional 7(1), 72 (2023)
K.J. Wang, A fractal modification of the unsteady korteweg-de vries model and its generalized fractal variational principle and diverse exact solutions. Fractals 30(9), 2250192 (2022)
A. Yusuf, T.A. Sulaiman, E.M. Khalil et al., Construction of multi-wave complexiton solutions of the Kadomtsev-Petviashvili equation via two efficient analyzing techniques. Results Phys 21, 103775 (2021)
K.J. Wang, Variational principle and diverse wave structures of the modified Benjamin-Bona-Mahony equation arising in the optical illusions field. Axioms 11(9), 445 (2022)
S.T.R. Rizvi, A.R. Seadawy, I. Ali et al., Chirp-free optical dromions for the presence of higher order spatio-temporal dispersions and absence of self-phase modulation in birefringent fibers. Mod. Phys. Lett. B 34(35), 2050399 (2020)
F. Zou, K.J. Wang, J.H. Liu, Abundant optical solitons of the (2+1)-dimensional Biswas-Milovic equation arising in optical fiber. Optik 252, 168510 (2022)
J. Yu, Y. Sun, Exact traveling wave solutions to the (2+1)-dimensional Biswas-Milovic equations. Optik 149, 378–383 (2017)
Z. Krpnar, Biswas-Milovic model and its optical solitons. Centr. Eur. Symp. Thermophys. 2116(1), 240004 (2019)
Q. Zhou, M. Ekici, A. Sonmezoglu et al., Optical solitons with Biswas-Milovic equation by extended G’/G-expansion method. Optik 127(16), 6277–6290 (2016)
S.T.R. Rizvi, K. Ali, M. Ahmad, Optical solitons for Biswas-Milovic equation by new extended auxiliary equation method. Optik 204, 164181 (2020)
H. Jafari, A. Sooraki, C.M. Khalique, Dark solitons of the Biswas-Milovic equation by the first integral method. Optik 124(19), 3929–3932 (2013)
M. Cinar, I. Onder, A. Secer et al., Optical solitons of the (2+1)-dimensional Biswas-Milovic equation using modified extended tanh-function method. Optik 245, 167631 (2021)
B. Kilic, M. Inc, Optical solitons for the Schrödinger-Hirota equation with power law nonlinearity by the Bäcklund transformation. Optik 138, 64–67 (2017)
M. Wang, Y. Wang, A new Bäcklund transformation and multi-soliton solutions to the KdV equation with general variable coefficients. Phys. Lett. A 287(3–4), 211–216 (2001)
M. Wang, Exact solutions for a compound KdV-Burgers equation. Phys. Lett. A 213(5–6), 279–287 (1996)
K.J. Wang, Bäcklund transformation and diverse exact explicit solutions of the fractal combined KdV-mKdV equation. Fractals 30(9), 2250189 (2022)
L. Tian, J. Yin, Stability of multi-compacton solutions and Backlund transformation in K (m, n, 1). Chaos Solitons Fractals 23(1), 159–169 (2005)
K.J. Wang, A fast insight into the optical solitons of the generalized third-order nonlinear Schrödinger’s equation. Results Phys. 40, 105872 (2022)
Acknowledgements
This work is supported by the Key Programs of Universities in Henan Province of China (22A140006), the Fundamental Research Funds for the Universities of Henan Province (NSFRF210324), Program of Henan Polytechnic University (B2018-40), the Innovative Scientists and Technicians Team of Henan Provincial High Education (21IRTSTHN016).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
This work does not have any conflicts of interest.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Wang, KJ., Liu, JH. Diverse optical solitons to the nonlinear Schrödinger equation via two novel techniques. Eur. Phys. J. Plus 138, 74 (2023). https://doi.org/10.1140/epjp/s13360-023-03710-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-023-03710-1