Abstract
In this work, the generalized unstable nonlinear Schrödinger equation is examined, which is used to predict the temporal evolution of disturbances in marginally stable or unstable media. First, we construct Lie symmetries and then, using corresponding transformations, we reduce the governing equation to a couple of ordinary differential equations. These coupled equations are solved and establish the invariant solutions, some of which are presented through graphs. Second, the dynamical behavior of the studied model is examined from various perspectives, including bifurcation, quasi-periodic, chaotic motion, and sensitivity analysis. Bifurcation analysis of the planar dynamical system is investigated at the equilibrium points of the system using bifurcation theory. After that, an external periodic perturbation term is introduced in the dynamical system, which is called the perturbed dynamical system. The quasi-periodic and chaotic motions of the perturbed dynamical system are identified through different chaos detecting tools including 3D phase portrait visualization, Poincare map, time series analysis, multistability analysis, bifurcation diagram and Lyapunov exponents. Using these tools, we observe that a perturbed dynamical system deviates from the regular patterns and exhibits chaotic behavior. Further, the sensitivity analysis is examined at three different initial conditions, and it is noticed that the given model is highly sensitive, as it changes significantly with even small variations in the initial condition. The reported results are novel, fascinating, and theoretically useful for understanding temporal evolution of disturbances in marginally stable or unstable media. Overall, understanding the dynamical behavior of systems and processes is crucial for making predictions, designing interventions, and developing new technologies.
Similar content being viewed by others
Availability of data and materials
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
Ai, W., Shi, Z., Liu, D.: Bifurcation analysis method of nonlinear traffic phenomena. Int. J. Mod. Phys. C 26(10), 1550111 (2015)
Akinyemi, L.: q-Homotopy analysis method for solving the seventh-order time-fractional Lax’s Korteweg-de Vries and Sawada-Kotera equations. Comput. Appl. Math. 38(4), 1–22 (2019)
Alhami, R., Alquran, M.: Extracted different types of optical lumps and breathers to the new generalized stochastic potential-KdV equation via using the Cole-Hopf transformation and Hirota bilinear method. Opt. Quantum Electron. 54(9), 1–2 (2022)
Alharbi, Y.F., Abdelrahman, M.A., Sohaly, M.A., Inc, M.: Stochastic treatment of the solutions for the resonant nonlinear Schrödinger equation with spatio-temporal dispersions and inter-modal using beta distribution. Eur. Phys. J. Plus 135(4), 1–4 (2020)
Ali, M., Alquran, M., Salman, O.B.: A variety of new periodic solutions to the damped (2+ 1)-dimensional Schrodinger equation via the novel modified rational sine-cosine functions and the extended tanh-coth expansion methods. Results Phys. 1(37), 105462 (2022)
Arora, G., Rani, R., Emadifar, H.: Numerical solutions of nonlinear Schrödinger equation with applications in optical fiber communication. Optik 266, 169661 (2022)
Cazenave, T., Lions, P.L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85(4), 549–61 (1982)
Centenera, M.M., Hickey, T.E., Jindal, S., Ryan, N.K., Ravindranathan, P., Mohammed, H., Robinson, J.L., Schiewer, M.J., Ma, S., Kapur, P., Sutherland, P.D.: A patient-derived explant (PDE) model of hormone-dependent cancer. Mol. Oncol. 12(9), 1608–22 (2018)
Chen, C.: Singular solitons of Biswas–Arshed equation by the modified simple equation method. Optik 184, 412–20 (2019)
Feng, B., Zhang, H.: Stability of standing waves for the fractional Schrödinger-Hartree equation. J. Math. Anal. Appl. 460(1), 352–64 (2018)
Gao, P., Chen, Y., Hou, L.: Bifurcation analysis for a simple dual-rotor system with nonlinear intershaft bearing based on the singularity method. Shock Vib. 2020 (2020). https://www.hindawi.com/journals/sv/2020/7820635/
Gepreel, K.A.: Exact soliton solutions for nonlinear perturbed Schrödinger equations with nonlinear optical media. Appl. Sci. 10(24), 8929 (2020)
Jhangeer, A., Raza, N., Rezazadeh, H., Seadawy, A.: Nonlinear self-adjointness, conserved quantities, bifurcation analysis and travelling wave solutions of a family of long-wave unstable lubrication model. Pramana 94(1), 1–9 (2020)
Jornet, M.: Modeling of Allee effect in biofilm formation via the stochastic bistable Allen-Cahn partial differential equation. Stoch. Anal. Appl. 39(1), 22–32 (2021)
Kaplan, M., Ozer, M.N.: Multiple-soliton solutions and analytical solutions to a nonlinear evolution equation. Opt. Quantum Electron. 50(1), 1 (2018)
Khater, M.M., Anwar, S., Tariq, K.U., Mohamed, M.S.: Some optical soliton solutions to the perturbed nonlinear Schrödinger equation by modified Khater method. AIP Adv. 11(2), 025130 (2021)
Kumar, S., Malik, S., Rezazadeh, H., Akinyemi, L.: The integrable Boussinesq equation and it’s breather, lump and soliton solutions. Nonlinear Dyn. 1–4 (2022). https://link.springer.com/article/10.1007/s11071-021-07076-w
Lei, S.: An analytical solution for steady flow into a Ttonnel. Groundwater 37(1), 23–6 (1999)
Li, L., Duan, C., Yu, F.: An improved Hirota bilinear method and new application for a nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation. Phys. Lett. A 383(14), 1578–82 (2019)
Liu, H., Yang, H., Liu, N., Yang, L.: Bifurcation and chaos analysis of tumor growth. Int. J. Biomath. 15, 2250039 (2022). https://www.worldscientific.com/doi/abs/10.1142/S1793524522500395
Liu, H.Z.: A modification to the first integral method and its applications. Appl. Math. Comput. 419, 126855 (2022)
Liu, X.Z., Yu, J.: A nonlocal nonlinear Schrödinger equation derived from a two-layer fluid model. Nonlinear Dyn. 96(3), 2103–2114 (2019)
Lu, D., Seadawy, A., Arshad, M.: Applications of extended simple equation method on unstable nonlinear Schrödinger equations. Optik 140, 136–44 (2017)
Ma, W., Shan, Y., Wang, B., Zhou, S., Wang, C.: Analytical solution for torsional vibration of an end-bearing pile in nonhomogeneous unsaturated soil. J. Build. Eng. 57, 104863 (2022)
Malik, S., Kumar, S., Das, A.: A (2+ 1)-dimensional combined KdV–mKdV equation: integrability, stability analysis and soliton solutions. Nonlinear Dyn. 1–3 (2022). https://link.springer.com/article/10.1007/s11071-021-07075-x
Malik, S., Hashemi, M.S., Kumar, S., Rezazadeh, H., Mahmoud, W., Osman, M.S.: Application of new Kudryashov method to various nonlinear partial differential equations. Opt. Quantum Electron. 55(1), 8 (2023)
Manukure, S., Booker, T.: A short overview of solitons and applications. Partial Differ. Equ. App. Math. 4, 100140 (2021)
Moitsheki, R.J., Makinde, O.D.: Classical Lie point symmetry analysis of nonlinear diffusion equations describing thermal energy storage. Appl. Math. Comput. 216(1), 251–60 (2010)
Nakkeeran, K.: Bright and dark optical solitons in fiber media with higher-order effects. Chaos Solitons Fractals 13(4), 673–9 (2002)
Omote, M., Sakagami, M., Sasaki, R., Yamanaka, I.: Solvable quantum field theories and polynomial conserved quantities for the quantum nonlinear Schrödinger equation. Phys. Rev. D 35(8), 2423 (1987)
Özer, AB., Akin, E.: Tools for detecting chaos. SA Fen Bilimleri Enstits Dergisi. 9, 60-64 (2005). https://physlab.lums.edu.pk/images/1/1f/Reference4.pdf
Pandir, Y., Ekin, A.: Dynamics of combined soliton solutions of unstable nonlinear Schrodinger equation with new version of the trial equation method. Chin. J. Phys. 67, 534–43 (2020)
Radha, B., Duraisamy, C.: The homogeneous balance method and its applications for finding the exact solutions for nonlinear equations. J. Ambient Intell. Humaniz. Comput. 12(6), 6591–7 (2021)
Rafiq, M.H., Jhangeer, A., Raza, N.: 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)
Raza, N., Ullah, M.A.: A comparative study of heat transfer analysis of fractional Maxwell fluid by using Caputo and Caputo-Fabrizio derivatives. Canad. J. Phys. 98(1), 89–101 (2020)
Raza, N., Rafiq, M.H., Kaplan, M., Kumar, S., Chu, Y.M.: The unified method for abundant soliton solutions of local time fractional nonlinear evolution equations. Results Phys. 22, 103979 (2021)
Rehman, H.U., Ullah, N., Imran, M.A.: Optical solitons of Biswas–Arshed equation in birefringent fibers using extended direct algebraic method. Optik 226, 165378 (2021)
Rezazadeh, H., Neirameh, A., Eslami, M., Bekir, A., Korkmaz, A.: A sub-equation method for solving the cubic-quartic NLSE with the Kerr law nonlinearity. Mod. Phys. Lett. B. 33(18), 1950197 (2019)
Rizvi, S.T., Seadawy, A.R., Mustafa, B., Ali, K., Ashraf, R.: Propagation of chirped periodic and solitary waves for the coupled nonlinear Schrödinger equation in two core optical fibers with parabolic law with weak non-local nonlinearity. Opt. Quantum Electron. 54(9), 1–46 (2022)
Saha, A.: Bifurcation, periodic and chaotic motions of the modified equal width-Burgers (MEW-Burgers) equation with external periodic perturbation. Nonlinear Dyn 87(4), 2193–201 (2017)
Salas, A.H., El-Tantawy, S.A., Castillo, H.J.E.: The hybrid finite difference and moving boundary methods for solving a linear damped nonlinear Schrödinger equation to model rogue waves and breathers in plasma physics. Math. Probl. Eng. 2020, 1–1 (2020)
Saliou, Y., Abbagari, S., Houwe, A., Osman, M.S., Yamigno, D.S., Crepin, K.T., Inc, M.: W-shape bright and several other solutions to the (3+ 1)-dimensional nonlinear evolution equations. Mod. Phys. Lett. B 35(30), 2150468 (2021)
Samina, S., Jhangeer, A., Chen, Z.: Bifurcation, chaotic and multistability analysis of the (2+1)-dimensional elliptic nonlinear Schrödinger equation with external perturbation. Waves Random Complex Media. 1–25 (2022). https://www.tandfonline.com/doi/full/10.1080/17455030.2022.2121010
Sgura, I., Lawless, A.S., Bozzini, B.: Parameter estimation for a morphochemical reaction-diffusion model of electrochemical pattern formation. Inverse Probl. Sci. Eng. 27(5), 618–47 (2019)
Shang, Y., Huang, Y., Yuan, W.: The extended hyperbolic functions method and new exact solutions to the Zakharov equations. Appl. Math. Comput. 200(1), 110–22 (2008)
Shi, L., Zhou, X.: Exact solutions of a coupled space-time fractional nonlinear Schrödinger type equation in quantum mechanics. Results Phys. 42, 105967 (2022)
Sprott, J.C.: A proposed standard for the publication of new chaotic systems. Int. J. Bifurc. Chaos 21(09), 2391–94 (2011)
Tala-Tebue, E., Djoufack, Z.I., Fendzi-Donfack, E., Kenfack-Jiotsa, A., Kofané, T.C.: Exact solutions of the unstable nonlinear Schrödinger equation with the new Jacobi elliptic function rational expansion method and the exponential rational function method. Optik 127(23), 11124–30 (2016)
Wazwaz, A.M.: Bright and dark optical solitons for (2+ 1)-dimensional Schrödinger (NLS) equations in the anomalous dispersion regimes and the normal dispersive regimes. Optik 192, 162948 (2019)
Wroblewski, M.: Nonlinear Schrödinger approach to European option pricing. Open Phys. 15(1), 280–291 (2017)
Yildirim, Y.: Optical solitons with Biswas–Arshed equation by F-expansion method. Optik 227, 165788 (2021)
Zhang, S.: A generalized auxiliary equation method and its application to the (2+ 1)-dimensional KdV equations. Appl. Math. Comput. 188(1), 1–6 (2007)
Zobeiry, N., Humfeld, K.D.: A physics-informed machine learning approach for solving heat transfer equation in advanced manufacturing and engineering applications. Eng. Appl. Artif. Intell. 101, 104232 (2021)
Zubair, A., Raza, N., Mirzazadeh, M., Liu, W., Zhou, Q.: Analytic study on optical solitons in parity-time-symmetric mixed linear and nonlinear modulation lattices with non-Kerr nonlinearities. Optik 173, 249–62 (2018)
Funding
There is no funding source.
Author information
Authors and Affiliations
Contributions
All authors contributed equally in preparation, drafting, editing and reviewing the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
Authors declare that they have not any competing interest of personal financial nature.
Ethical approval and consent to participate
The authors declare that there is no conflict with publication ethics.
Consent for publication
The authors declare that there is no conflict with publication of this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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
Rafiq, M.H., Raza, N. & Jhangeer, A. Nonlinear dynamics of the generalized unstable nonlinear Schrödinger equation: a graphical perspective. Opt Quant Electron 55, 628 (2023). https://doi.org/10.1007/s11082-023-04904-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-023-04904-8