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Dynamical properties and travelling wave analysis of Rangwala–Rao equation by complete discrimination system for polynomials

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Abstract

In this research, we study a variety of solutions like periodic, exponential, rational, hyperbolic and Jacobian elliptic function solutions of Rangwala–Rao equation (RRE) by employing complete discrimination system for polynomial (CDSP) technique. We deduce the relevant travelling wave system from the original equation using the travelling wave transformation and generate a conserved quantity, the Hamiltonian, from it. With the aid of qualitative theory of differential equation and bifurcation theory of planar dynamical systems a variety of phase portraits to the relevant travelling wave system of RRE are explored. The CDSP approach is not only useful for generating soliton solutions but also for carrying out qualitative analysis. Furthermore, 3D and 2D graphs are displayed for some of the soliton solutions of RRE.

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References

  • Akinyemi, L., Mirzazadeh, M., Hosseini, K.: Solitons and other solutions of perturbed nonlinear Biswas–Milovic equation with Kudryashovs law of refractive index. Nonlinear Anal. Model. Control 27, 1–17 (2022)

    Article  MathSciNet  Google Scholar 

  • Aly, R., Seadawy, Muhammad Arshad, Dianchen, Lu.: The weakly nonlinear wave propagation theory for the Kelvin–Helmholtz instability in magnetohydrodynamics flows. Chaos, Solitons Fractals 139, 110141 (2020)

    Article  MathSciNet  Google Scholar 

  • Asjad, M.I., Faridi, W.A., Alhazmi, S.E., Hussanan, A.: The modulation instability analysis and generalized fractional propagating patterns of the Peyrard–Bishop DNA dynamical equation. Opt. Quantum Electron. 55(3), 1–34 (2023)

    Article  Google Scholar 

  • Bo, W.B., Wang, R.R., Fang, Y., Wang, Y.Y., Dai, C.Q.: Prediction and dynamical evolution of multipole soliton families in fractional Schrödinger equation with the PT-symmetric potential and saturable nonlinearity. Nonlinear Dyn. 111(2), 1577–88 (2023)

    Article  Google Scholar 

  • Cao, D.: The classification of the single traveling wave solutions to the time-fraction Gardner equation. Chin. J. Phys. 59, 379–92 (2019)

    Article  MathSciNet  Google Scholar 

  • Chen, Y.X., Xiao, X.: Vector soliton pairs for a coupled nonautonomous NLS model with partially nonlocal coupled nonlinearities under the external potentials. Nonlinear Dyn. 109(3), 2003–12 (2022)

    Article  Google Scholar 

  • Chen, S., Liu, Y., Wei, L., Guan, B.: Exact solutions to fractional Drinfel’d–Sokolov–Wilson equations. Chin. J. Phys. 56(2), 708–20 (2018)

    Article  MathSciNet  Google Scholar 

  • Eslami, M., Rezazadeh, H.: The first integral method for Wu-Zhang system with conformable time-fractional derivative. Calcolo 53(3), 475–485 (2016)

    Article  MathSciNet  Google Scholar 

  • Fang, Y., Wu, G.Z., Wang, Y.Y., Dai, C.Q.: Data-driven femtosecond optical soliton excitations and parameters discovery of the high-order NLSE using the PINN. Nonlinear Dyn. 105(1), 603–16 (2021)

    Article  Google Scholar 

  • Geng, X., Li, R., Xue, B.: A vector general nonlinear Schrödinger equation with \((m+n)\)-components. J. Nonlinear Sci. 30(3), 991–1013 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  • Geng, K.L., Zhu, B.W., Cao, Q.H., Dai, C.Q., Wang, Y.Y.: Nondegenerate soliton dynamics of nonlocal nonlinear Schrödinger equation. Nonlinear Dyn. 111(17), 16483–96 (2023)

    Article  Google Scholar 

  • Hermann, J., Schätzle, Z., Noé, F.: Deep-neural-network solution of the electronic Schrödinger equation. Nat. Chem. 12(10), 891–897 (2020)

    Article  Google Scholar 

  • Iqbal, Mujahid, Seadawy, Aly R., Dianchen, Lu.: Applications of nonlinear longitudinal wave equation in a magneto-electro-elastic circular rod and new solitary wave solutions. Mod. Phys. Lett. B 33(18), 1950210 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  • Jhangeer, A., Rezazadeh, H., Seadawy, A.: A study of travelling, periodic, quasiperiodic and chaotic structures of perturbed Fokas–Lenells model. Pramana 95, 1–11 (2021)

    Article  ADS  Google Scholar 

  • Kai, Y., Chen, S., Zhang, K., Yin, Z.: A study of the shallow water waves with some Boussinesq-type equations. In: Waves Random Complex Media, pp. 1–18 (2021)

  • Kai, Y.: The classification of the single travelling wave solutions to the variant Boussinesq equations. Pramana J. Phys. 87(4), 59 (2016)

    Article  ADS  Google Scholar 

  • Kai, Y., Huang, L.: Dynamic properties, Gaussian soliton and chaotic behaviors of general Degasperis–Procesi model. Nonlinear Dyn. 111(9), 8687–700 (2023)

    Article  MathSciNet  Google Scholar 

  • Kai, Y., Chen, S., Zheng, B., Zhang, K., Yang, N., Xu, W.: Qualitative and quantitative analysis of nonlinear dynamics by the complete discrimination system for polynomial method. Chaos Solitons Fractals 141, 110314 (2020)

    Article  MathSciNet  Google Scholar 

  • Khater, A.H., Callebaut, D.K., Malfliet, W., Seadawy, A.R.: Nonlinear dispersive Rayleigh–Taylor instabilities in magnetohydrodynamic flows. Phys. Scr. 64, 533–547 (2001)

    Article  ADS  Google Scholar 

  • Kudryashov, N.A.: Almost general solution of the reduced higher-order nonlinear Schrödinger equation. Optik 230, 166347 (2021)

    Article  ADS  Google Scholar 

  • Liu, X.H.: Exact solitary wave solutions of the Rangwala–Rao equation. In: 2012 2nd International Conference on Uncertainty Reasoning and Knowledge Engineering, pp. 175-178. IEEE (2012)

  • Manzhos, S.: Machine learning for the solution of the Schrödinger equation. Mach. Learn. Sci. Technol. 1(1), 013002 (2020)

    Article  Google Scholar 

  • Özkan, Y.S., Yasar, E., Seadawy, A.R.: A third-order nonlinear Schrödinger equation: the exact solutions, group-invariant solutions and conservation laws. J. Taibah Univ. Sci. 14(1), 585–597 (2020)

    Article  Google Scholar 

  • Rezazadeh, H.: New solitons solutions of the complex Ginzburg–Landau equation with Kerr law nonlinearity. Optik 167, 218–227 (2018)

    Article  ADS  Google Scholar 

  • Rizvi, S.T.R., Abbas, S.O., Ali, K.: Optical solitons for non-Kerr law nonlinear Schrödinger equation with third and fourth order dispersions. Chin. J. Phys. 60, 133–140 (2019)

    Article  Google Scholar 

  • Russell, P.: Photonic crystal fibers. Science 299(5605), 358–62 (2003)

    Article  ADS  Google Scholar 

  • Seadawy, A.R.: Stability analysis for Zakharov–Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma. Comput. Math. Appl. 67, 172–180 (2014)

    Article  MathSciNet  Google Scholar 

  • Seadawy, Aly R.: Approximation solutions of derivative nonlinear Schrodinger equation with computational applications by variational method. Eur. Phys. J. Plus 130(182), 1–10 (2015)

    Google Scholar 

  • Seadawy, A.R.: Stability analysis solutions for nonlinear three-dimensional modified Korteweg-de Vries–Zakharov–Kuznetsov equation in a magnetized electron-positron plasma. Phys. A Stat. Mech. Appl. Phys. A 455, 44–51 (2016)

    Article  MathSciNet  Google Scholar 

  • Seadawy, Aly R., Alsaedi, Bayan: Contraction of variational principle and optical soliton solutions for two models of nonlinear Schrodinger equation with polynomial law nonlinearity. AIMS Math. 9(3), 6336–6367 (2024)

    Article  MathSciNet  Google Scholar 

  • Seadawy, Aly R., Alsaedi, Bayan A.: Soliton solutions of nonlinear Schrödinger dynamical equation with exotic law nonlinearity by variational principle method. Opt. Quantum Electron. 56, 700 (2024)

    Article  Google Scholar 

  • Seadawy, Aly R., Rizvi, Syed T. R., Ali, Ijaz, Younis, Muhammad, Kashif Ali, M.M., Makhlouf, Ali Althobaiti: Conservation laws, optical molecules, modulation instability and Painlevè analysis for the Chen–Lee–Liu model. Opt. Quantum Electron. 53, 172 (2021)

    Article  Google Scholar 

  • Shi, L.C.: Exact travelling wave solutions for \((1+1)\)-dimensional dispersive long wave equation. Chin. J. 14(9), 1710–1715 (2005)

    Google Scholar 

  • Singh, S.S.: Exact solutions of Kundu–Eckhaus equation and Rangwala–Rao equation by reduction to Lienard equation. J. Math. Phys. 2016, 11 (2016)

    Google Scholar 

  • Wang, Jun, Shehzad, Khurrem, Seadawy, Aly R., Arshad, Muhammad, Asmat, Farwa: 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)

    Article  Google Scholar 

  • Wei, T., Guan, B., Chen, S., Li, Y., Cao, M., Meng, L., Lin, X.: Wave patterns and dynamical properties of optical propagation by a higher order nonlinear Schrödinger equation. Results Phys. 46, 106283 (2023)

    Article  Google Scholar 

  • Wen, X.K., Jiang, J.H., Liu, W., Dia, C.Q.: Abundant vector soliton prediction and model parameter discovery of the couples mixed derivative nonlinear Schrödinger equation. Nonlinear Dyn. 111, 13343–13355 (2023)

    Article  Google Scholar 

  • Werther, M., Choudhury, S.L., Großmann, F.: Coherent state based solutions of the time-dependent Schrödinger equation: hierarchy of approximations to the variational principle. Int. Rev. Phys. Chem. 40(1), 81–125 (2021)

    Article  Google Scholar 

  • Xu, S.Y., Zhou, Q., Liu, W.: Prediction of soliton evolution and equation parameters for NLS-MB equation based on the phPINN algorithm. Nonlinear Dyn. 111(19), 18401–17 (2023)

    Article  Google Scholar 

  • Yokus, A.: Construction of different types of traveling wave solutions of the relativistic wave equation associated with the Schrödinger equation. Math. Model. Numer. Simul. Appl. 1(1), 24–31 (2021)

    Google Scholar 

  • Zhang, J.L., Wang, M.L.: Exact solutions to a class of nonlinear Schrödinger-type equations. Pramana 67(6), 1011–1022 (2006)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through Project Number (TU-DSPP-2024-87).

Funding

This research was funded by Taif University, Saudi Arabia, Project No (TU-DSPP-2024-87).

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Correspondence to Aly. R. Seadawy.

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Ali, K., Seadawy, A.R., Rizvi, S.T.R. et al. Dynamical properties and travelling wave analysis of Rangwala–Rao equation by complete discrimination system for polynomials. Opt Quant Electron 56, 1081 (2024). https://doi.org/10.1007/s11082-024-06894-7

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