Abstract
Bifurcation analysis is a powerful method for investigating the steady-state nonlinear dynamics of systems. There are software programmes that allow for the numerical continuation of steady-state solutions while the system’s parameters are changed. The aim of this manuscript is to deal a mixed derivative nonlinear Schrödinger’s equation (MD-NLSE), \(\alpha \)-helix NLSE and Zoomeron model utilising the bifurcation theory technique of dynamical systems. A mathematical method called bifurcation analysis is used to examine how a system behaves when a parameter is changed. It is used to examine how the behaviour of the system changes as one or more parameters are altered and to find critical values of the parameters that cause appreciable changes in the behaviour of the system. Bifurcations are places in parameter space when the system’s qualitative behaviour abruptly shifts. A system could, for instance, go from having a stable equilibrium to oscillating between two states when a parameter is increased. To investigate these changes and comprehend how the system’s behaviour would alter if the parameter is further altered, bifurcation analysis is performed. An example of a phase portrait is a plot of a dynamic system’s phase space, which graphically depicts the behaviour of the system. A two-dimensional depiction of the system’s variables on the x and y axes is a common way to depict the phase space, which is the space of all conceivable states of the system. The paths of the system are shown as curves or lines in the phase space in a phase picture. These paths show how the system has changed over time and may be used to examine the stability and behaviour of the system. Phase portraits are often employed in the study of the behaviour of complex systems in physics, engineering, and mathematics. Numerous systems, including mechanical systems, electrical circuits, chemical processes, and biological systems, may be studied using them. Researchers can learn more about a system’s dynamics and forecast how it will behave in various situations by looking at its phase picture.
Similar content being viewed by others
Data availability
Not applicable.
References
Abbagari, S., Houwe, A., Akinyemi, L., Saliou, Y., Bouetou, T.B.: Modulation instability gain and discrete soliton interaction in gyrotropic molecular chain. Chaos, Solit. Fract. 160, 112255 (2022). https://doi.org/10.1016/j.chaos.2022.112255
Ahmad, H., Alam, N., Omri, M.: New computational results for a prototype of an excitable system. Res. Phys. (2021). https://doi.org/10.1016/j.rinp.2021.104666
Akinyemi, L., Mirzazadeh, M., Hosseini, K.: Solitons and other solutions of perturbed nonlinear Biswas-Milovic equation with Kudryashov’s law of refractive index. Nonlinear Anal.: Modell. Control 27, 1–17 (2022). https://doi.org/10.15388/namc.2022.27.26374
Aly, R.: Seadawy, Stability analysis solutions for nonlinear three-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation in a magnetized electron-positron plasma, Physica A: Statistical Mechanics and its Applications. Phys. A 455, 44–51 (2016)
Aly, R., Seadawy, M.I., Lu, D.: Applications of propagation of long-wave with dissipation and dispersion in nonlinear media via solitary wave solutions of generalized Kadomtsive-Petviashvili modified equal width dynamical equation. Comput. Math. Appl. 78, 3620–3632 (2019)
Aly, R., Seadawy, M.A., Lu, D.: The weakly nonlinear wave propagation theory for the Kelvin-Helmholtz instability in magnetohydrodynamics flows. Chaos Solit. Fract. 139, 110141 (2020)
Aziz, N., Ali, K., Seadawy, A.R., Bashir, A., Rizvi, S.T.R.: Discussion on couple of nonlinear models for lie symmetry analysis, self adjointees, conservation laws and soliton solutions. Opt. Quant. Electron. 55, 201 (2023)
Bashir, A., Seadawy, A.R., Ahmed, S., Rizvi, S.T.R.: The Weierstrass and Jacobi elliptic solutions along with multiwave, homoclinic breather, kink-periodic-cross rational and other solitary wave solutions to Fornberg Whitham equation. Chaos, Solit. Fract. 163, 112538 (2022)
Batool, T., Seadawy, A.R., Rizvi, S.T.R., Naqvi, S.K.: Optical multi-wave, \(M\)-shaped rational solution, homoclinic breather, periodic cross-kink and various rational solutions with interactions for Radhakrishnan-Kundu-Lakshmanan dynamical model. J. Nonlinear Opt. Phys. Mater. 32(2), 2350015 (2023)
Bhrawy, A.H., Alshaery, A.A., Hilal, E.M., Milovic, D., Moraru, L., Savescu, M., Biswas, A.: Optical solitons with polynomial and triple-power law nonlinearities and spatio-temporal dispersion. Proc. Roman. Acad. Ser. A 15, 235–240 (2014)
Biswas, A., Moran, A., Milovic, D., Majid, F., Biswas, K.C.: An exact solution for the modified nonlinear Schrödinger’s equation for Davydov solitons in \(\alpha \)-helix proteins. Math. Biosci. 227, 68–71 (2010)
Biswas, A., Song, M., Triki, H., Kara, A.H., Ahmed, B.S., Strong, A., Hama, A.: Solitons, shock waves, conservation laws and bifurcation analysis of Boussinesq equation with power law nonlinearity and dual dispersion. Appl. Math. Inform. Sci. 8(3), 949–957 (2014)
Çelik, N., Seadawy, A.R., Özkan, Y.S., Yaşar, E.: A model of solitary waves in a nonlinear elastic circular rod: abundant different type exact solutions and conservation laws. Chaos, Solit. Fract. 143, 110486 (2021)
Chabchoub, A., Grimshaw, R.H.J.: The hydrodynamic nonlinear Schrödinger equation: space and time. Fluids 1(3), 23 (2016)
Eslami, M., Rezazadeh, H.: The first integral method for Wu-Zhang system with conformable time-fractional derivative. Calcolo 53(3), 475–485 (2016). https://doi.org/10.1007/s10092-015-0158-8
Farah, N., Seadawy, A.R., Rizvi, S.T.R., Ahmad, S.: Butterfly, \(S\) and \(W\)-shaped, parabolic, and other soliton solutions to the improved perturbed nonlinear Schrödinger equation. Opt. Quant. Electron. 55, 99 (2023)
Faridi, W.A., Asjad, M.I., Toseef, M., Amjad, T.: Analysis of propagating wave structures of the cold bosonic atoms in a zig-zag optical lattice via comparison with two different analytical techniques. Opt. Quant. Electron. 54, 773 (2022)
Faridi, W.A., Asjad, M.I., Jarad, F.: The fractional wave propagation, dynamical investigation, and sensitive visualization of the continuum isotropic bi-quadratic Heisenberg spin chain process. Res. Phys. 43, 106039 (2022)
Faridi, W.A., Asjad, M.I., Jarad, F.: Non-linear soliton solutions of perturbed Chen-Lee-Liu model by -model expansion approach. Opt. Quant. Electron. 54, 664 (2022)
Faridi, W.A., Bakar, M.A., Akgül, A., El-Rahman, M.A., El Din, S.M.: Exact fractional soliton solutions of thin-film ferroelectric material equation by analytical approaches. Alex. Eng. J. 78, 483–497 (2023)
Faridi, W.A., Asjad, M.I., Jhangeer, A., Yusuf, A., Sulaiman, T.A.: The weakly non-linear waves propagation for Kelvin-Helmholtz instability in the magnetohydrodynamics flow impelled by fractional theory. Opt. Quant. Electron. 55, 172 (2023)
Gao, W., Rezazadeh, H., Pinar, Z., Baskonus, H.M., Sarwar, S., Yel, G.: Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique. Opt. Quant. Electron. 52, 52 (2020)
Han, T., Li, Z., Zhang, X.: Bifurcation and new exact traveling wave solutions to time-space coupled fractional nonlinear Schrödinger equation. Phys. Lett. A 395, 127217 (2021)
Hosseini, K., Hinçal, E., Ilie, M.: Bifurcation analysis, chaotic behaviors, sensitivity analysis, and soliton solutions of a generalized Schrödinger equation. Nonlinear Dyn. 111, 17455–17462 (2023)
Izgi, Z.P.: Rogue waves and solitons of the generalized modified nonlinear Schrödinger equations. Math. Comput. Simul. 208, 535–549 (2023)
Jhangeer, A., Almusawa, H., Hussain, Z.: Bifurcation study and pattern formation analysis of a nonlinear dynamical system for chaotic behavior in traveling wave solution. Res. Phys. 37, 105492 (2022)
Karaaslanli, C. Ç.: Bifurcation Analysis and Its Applications, Intech Open Access Publisher, (2012)
Kopçasız, B., Yaşar, E.: Novel exact solutions and bifurcation analysis to dual-mode nonlinear Schrödinger equation. J. Ocean Eng. Sci. 213, 752–771 (2022). https://doi.org/10.1016/j.joes.2022.06.007
Li, Y., Shan, W. r., Shuai, T., Rao, K.: Bifurcation analysis and solutions of a higher-order nonlinear Schrödinger Equation. Math. Prob. Eng. 10 (2015)
Mezamo, N.C.T., Nana, V.B., Tchuimmo, F.W., Nanaa, L.: Stabilization of traveling waves on dissipative system near subcritical bifurcation through a combination of global and local feedback. Eur. Phys. J. Plus (2022). https://doi.org/10.1140/epjp/s13360-022-03352-9
Miyaji, T., Ohnishi, I., Tsutsumi, Y.: Bifurcation analysis to the Lugiato-Lefever equation in one space dimension. Physica D 239, 2066–2083 (2010)
Raza, N., Rafiq, M.H., Alrebdi, T.A., Abdel-Aty, A.-H.: New solitary waves, bifurcation and chaotic patterns of Coupled Nonlinear Schrodinger System arising in fibre optics. Opt. Quant. Electron. 55, 853 (2023)
Rezazadeh, H.: New solitons solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity. Optik 167, 218–227 (2018). https://doi.org/10.1016/j.ijleo.2018.04.0263
Rizvi, S. T. R., Seadawy, A. R., Naqvi, S. K., Abbas, S. O.: Study of mixed derivative nonlinear Schrödinger equation for rogue and lump waves, breathers and their interaction solutions with Kerr law. Opt. Quant. Electron. 55, 177 (2023)
Rizvi, S.T.R., Seadawy, A.R., Batool, T., Ali, K.: Several new analytical solutions for Davydov solitons in \(\alpha \)-helix proteins. Int. J. Mod. Phys. B 36(30), 2250213 (2022)
Rizvi, S.T.R., Seadawy, A.R., Ahmad, S., Ali, K.: Einsteiés vacuum field equation: lumps, manifold, periodic, generalized breathers, interaction and rogue wave solutions. Opt. Quant. Electron. 55, 181 (2023)
Rizvi, S.T.R., Seadawy, A.R., Naqvi, S.K., Abbas, S.O.: Study of mixed derivative nonlinear Schrödinger equation for rogue and lump waves, breathers and their interaction solution with Kerr law. Opt. Quant. Electron. 55, 177 (2023)
Seadawy, A.R., Iqbal, M., Lu, D.: Application of mathematical methods on the ion sound and Langmuir waves dynamical systems. The Pramana J. Phys. 93, 10 (2019)
Seadawy, A.R., Rizvi, S.T.R., Ahmed, S., Bashir, A.: Lump solutions, Kuznetsov-Ma breathers, rogue waves and interaction solutions for magneto electro-elastic circular rod. Chaos Solit. Fract. 163, 112563 (2022)
Seadawy, A.R., Rizvi, S.T.R., Akram, U., Naqvi, S.K.: Optical and analytical solitons to higher order non-Kerr nonlinear Schrödinger dynamical model. J. Geomet. Phys. 179, 104616 (2022)
Seadawy, A.R., Rizvi, S.T.R., Sohail, M., Ali, K.: Nonlinear model under anomalous dispersion regime: chirped periodic and solitary waves. Chaos, Solit. Fract. 163, 112558 (2022)
Seadawy, A.R., Rizvi, S.T.R., Ahmed, S., Batool, T.: Propagation of \(W\)-shaped and \(M\)-shaped solitons with mulit-peak interaction for ultrashort light pulse in fibers. Opt. Quant. Electron. 55, 221 (2023)
Seadawy, A.R., Ahmed, S., Rizvi, S.T.R., Nazar, K.: Applications for mixed Chen-Lee-Liu derivative nonlinear Schrödinger equation in water wave flumes and optical fibers. Opt. Quant. Electron. 55, 34 (2023)
Shah, K., Seadawy, A.R., Arfan, M.: Evaluation of one dimensional fuzzy fractional partial differential equations. Alex. Eng. J. 59, 3347–3353 (2020)
Shahein, R.A., El-Shehri, J.H.: Bifurcation analysis of dissipative rogue wave in electron-positron-ion plasma with relativistic ions and superthermal electrons. Chaos, Solit. Fract. 128, 114–122 (2019)
Ullah, M.S., Roshid, H.O., Ali, M.Z., Rezazadeh, H.: Kink and breather waves with and without singular solutions to the Zoomeron model. Res. Phys. 49, 106535 (2023)
Wang, J., Shehzad, K., Seadawy, A.R., Arshad, M., Asmat, F.: Dynamic study of multi-peak solitons and other wave solutions of new coupled KdV and new coupled Zakharov-Kuznetsov systems with their stability. J. Taibah Univ. Sci. 17(1), 2163872 (2023)
Younas, U., Seadawy, A.R., Younis, M., Rizvi, S.T.R.: Construction of analytical wave solutions to the conformable fractional dynamical system of ion sound and Langmuir waves. Waves Rand. Compl. Media 32(6), 2587–2605 (2022)
Zhou, Y., Cai, S., Liu, Q.: Bounded traveling waves of the (2+1)-dimensional Zoomeron equation. Math. Prob. Eng. (2015). https://doi.org/10.1155/2015/163597
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Conflict of Interest
The authors declare no conflict of interest.
Ethical approval
I hereby declare that this manuscript is the result of my independent creation under the reviewers’ comments. Except for the quoted contents, this manuscript does not contain any research achievements that have been published or written by other individuals or groups.
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
Rizvi, S.T.R., Seadawy, A.R., Naqvi, S.K. et al. Bifurcation analysis for mixed derivative nonlinear Schrödinger’s equation , \(\alpha \)-helix nonlinear Schrödinger’s equation and Zoomeron model. Opt Quant Electron 56, 452 (2024). https://doi.org/10.1007/s11082-023-06100-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-023-06100-0