Abstract
For exploring the behavior of complex two-dimensional systems, the idea of planar dynamical systems offers a potent and adaptable framework. It is simple for researchers to understand the dynamics of the system and acquire deeper insights into its behavior by visualizing trajectories in the (x,y) plane. Keeping this in mind, the complex dynamics of the nonlinear Schrödinger equation has not been explored using planar dynamical theory, in the literature. In this manuscript, the dynamical features and some new optical solitons are discussed for the considered NLSE. Using RK4 numerical technique, the dynamical features such as bifurcation, phase portraits, time series and sensitivity analysis are portrayed. Moreover, utilizing the traveling wave transformations and complete discriminant system method, some new optical solitary waves solutions are obtained which depends on trigonometric functions and Jacobi elliptic functions. These solutions are new and have not been reported in the literature for the considered equation.
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References
M. Ming, F. Rousset, N. Tzvetkov, Multi-solitons and related solutions for the water-waves system. SIAM J. Math. Anal. 47(1), 897–954 (2015)
P. Marin-Palomo, J.N. Kemal, M. Karpov, A. Kordts, J. Pfeifle, M.H.P. Pfeiffer, P. Trocha et al., Microresonator-based solitons for massively parallel coherent optical communications. Nature 546(7657), 274–279 (2017)
H. Nawaz, W. Masood, R. Jahangir, M. Siddiq, Interaction of Gardner solitons in plasmas: applications in the Saturn’s magnetosphere. Phys. Scr. 96(4), 045604 (2021)
A. Dakova, D. Dakova, Z. Andreeva, V. Slavchev, L. Kovachev, Mutual action of self-phase modulation and cross-phase modulation on the parametric four-photon mixing. Exact Analytical solutions in the form of Jacobi functions’’. Optik 194, 163024 (2019)
J. Zhang, H.-Q. Hao, Soliton solutions of the AB system via the Jacobi elliptic function expansion method. Optik 244, 167541 (2021)
E.M.E. Zayed, A.G. Al-Nowehy, M.E.M. Elshater, Solitons and other solutions to nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity using several different techniques. Eur. Phys. J. Plus 132, 1–14 (2017)
Z. Kasapeteva, A. Dakova, S. Krasteva, V. Slavchev, D. Dakova, L. Kovachev, A. Biswas, Bright solitons under the influence of third-order dispersion and self-steepening effect. Opt. Quantum Electron. 54(6), 352 (2022)
A.-M. Wazwaz, S.A. El-Tantawy, Optical Gaussons for nonlinear logarithmic Schrödinger equations via the variational iteration method. Optik 180, 414–418 (2019)
A.-M. Wazwaz, Bright and dark optical solitons of the (2+ 1)-dimensional perturbed nonlinear Schrödinger equation in nonlinear optical fibers. Optik 251, 168334 (2022)
A.-M. Wazwaz, W. Alhejaili, S.A. El-Tantawy, Bright and dark envelope optical solitons for a (2+ 1)-dimensional cubic nonlinear Schrödinger equation. Optik 265, 169525 (2022)
W.-X. Ma, Matrix integrable fourth-order nonlinear Schrödinger equations and their exact soliton solutions. Chin. Phys. Lett. 39(10), 100201 (2022)
D. Lu, A.R. Seadawy, J. Wang, M. Arshad, U. Farooq, Soliton solutions of the generalised third-order nonlinear Schrödinger equation by two mathematical methods and their stability. Pramana 93, 1–9 (2019)
S Ahmad, Salman, A Ullah, S Ahmad, A Akgül. “Bright, dark and hybrid multistrip optical soliton solutions of a non-linear Schrödinger equation using modified extended tanh technique with new Riccati solutions.” Opt. Quantum Electron. 55, no. 3: 236 (2023)
M. Ozisik, A. Secer, M. Bayram, On solitary wave solutions for the extended nonlinear Schrödinger equation via the modified F-expansion method. Opt. Quantum Electron. 55(3), 215 (2023)
S. Ahmad, A. Hameed, S. Ahmad, A. Ullah, M. Akbar, Stability analysis and some exact solutions of a particular equation from a family of a nonlinear Schrödinger equation with unrestricted dispersion and polynomial nonlinearity. Opt. Quantum Electron. 55(8), 666 (2023)
S.T.R. Rizvi, A.R. Seadawy, N. Farah, S. Ahmad, Application of Hirota operators for controlling soliton interactions for Bose-Einstien condensate and quintic derivative nonlinear Schrödinger equation. Chaos Solitons Fractals 159, 112128 (2022)
K.U. Tariq, A.R. Seadawy, A. Ahmed, Some novel solitary wave solutions to the generalized coupled nonlinear Schrödinger-Korteweg-de Vries equations. Res. Phys. 35, 105321 (2022)
A. Saha, S. Banerjee, Dynamical Systems and Nonlinear Waves in Plasmas (CRC Press, Florida, 2021)
P.N.V. Tu, Dynamical Systems: An Introduction with Applications in Economics and Biology (Springer Science & Business Media, Cham, 2012)
M. Karimi-Ghartemani, A.K. Ziarani, Periodic orbit analysis of two dynamical systems for electrical engineering applications. J. Eng. Math. 45, 135–154 (2003)
C. Xu, Q. Cui, Z. Liu, Y. Pan, X. Cui, W. Ou, M. Rahman, M. Farman, S. Ahmad, A. Zeb, Extended hybrid controller design of bifurcation in a delayed chemostat model. MATCH Commun. Math. Comput. Chem. 90(3), 609–648 (2023). https://doi.org/10.46793/match.90-3.609X
P.L. Li, Y.J. Lu, C.J. Xu, J. Ren, Insight into Hopf bifurcation and control methods in fractional order BAM neural networks incorporating symmetric structure and delay. Cogn. Comput. (2023). https://doi.org/10.1007/s12559-023-10155-2
D. Mu, C. Xu, Z. Liu, Y. Pang, Further insight into bifurcation and hybrid control tactics of a chlorine dioxide-iodine-malonic acid chemical reaction model incorporating delays. MATCH Commun. Math. Comput. Chem. 89(3), 529–566 (2023). https://doi.org/10.46793/match.89-3.529M
P. Li, R. Gao, X. Changjin, J. Shen, S. Ahmad, Y. Li, Exploring the impact of delay on Hopf bifurcation of a type of BAM neural network models concerning three nonidentical delays. Neural Process. Lett. (2022). https://doi.org/10.1007/s11063-022-11118-8
P. Li, X. Peng, X. Changjin, L. Han, S. Shi, Novel extended mixed controller design for bifurcation control of fractional-order Myc/E2F/miR-17-92 network model concerning delay. Math. Methods Appl. Sci. (2023). https://doi.org/10.1002/mma.9597
C. Xu, M. Liao, P. Li, L. Yao, Q. Qin, Y. Shang, Chaos control for a fractional-order Jerk system via time delay feedback controller and mixed controller. Fractal Fract. 5(4), 257 (2021). https://doi.org/10.3390/fractalfract5040257
Q.Z. He, P.F. Xia, C. Hu, B. Li, PUBLIC information, actual intervention and inflation expectations. Transf. Bus. Econ. 21(3C), 644–666 (2022)
Q. He, X. Zhang, P. Xia, C. Zhao, S. Li, A comparison research on dynamic characteristics of Chinese and American energy prices. J. Glob. Inf. Manag. (JGIM) 31(1), 1–16 (2023)
C. Xu, Z. Liu, P. Li, J. Yan, L. Yao, Bifurcation mechanism for fractional-order three-triangle multi-delayed neural networks. Neural Process. Lett. 55, 6125–6151 (2023). https://doi.org/10.1007/s11063-022-11130-y
C. Xu, D. Mu, Z. Liu, Y. Pang, M. Liao, P. Li, Bifurcation dynamics and control mechanism of a fractional-order delayed Brusselator chemical reaction model. MATCH Commun. Math. Comput. Chem. 89(1), 73–106 (2023). https://doi.org/10.46793/match.89-1.073X
B. Li, T. Zhang, C. Zhang, Investigation of financial bubble mathematical model under fractal-fractional Caputo derivative. Fractals 31(05), 1–13 (2023)
B. Li, Z. Eskandari, Dynamical analysis of a discrete-time SIR epidemic model. J. Frankl. Inst. 360(12), 7989–8007 (2023)
L. Tang, Bifurcation analysis and multiple solitons in birefringent fibers with coupled Schrödinger-Hirota equation. Chaos Solitons Fractals 161, 112383 (2022)
Y. Song, B. Yang, Z. Wang, Bifurcations and exact solutions of a new (3+ 1)-dimensional Kadomtsev-Petviashvili equation. Phys. Lett. A 461, 128647 (2023)
M.H. Rafiq, A. Jhangeer, N. Raza, The analysis of solitonic, supernonlinear, periodic, quasiperiodic, bifurcation and chaotic patterns of perturbed Gerdjikov-Ivanov model with full nonlinearity’. Commun. Nonlinear Sci. Numer. Simul. 116, 106818 (2023)
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Khan, A., Saifullah, S., Ahmad, S. et al. Dynamical properties and new optical soliton solutions of a generalized nonlinear Schrödinger equation. Eur. Phys. J. Plus 138, 1059 (2023). https://doi.org/10.1140/epjp/s13360-023-04697-5
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DOI: https://doi.org/10.1140/epjp/s13360-023-04697-5