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Construction of breather solutions and N-soliton for the higher order dimensional Caudrey–Dodd–Gibbon–Sawada–Kotera equation arising from wave patterns

  • Hajar F. Ismael , Aly Seadawy EMAIL logo and Hasan Bulut
Published/Copyright: March 11, 2021
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International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 24 Issue 1

Abstract

In this research, we explore the dynamics of Caudrey–Dodd–Gibbon–Sawada–Kotera equations in (1 + 1)-dimension, such as N-soliton, and breather solutions. First, a logarithmic variable transform based on the Hirota bilinear method is defined, and then one, two, three and N-soliton solutions are constructed. A breather solution to the equation is also retrieved via N-soliton solutions. All the solutions that have been obtained are novel and plugged into the equation to guarantee their existence. 2-D, 3-D, contour plot and density plot are also presented.

Keywords: breather solutions; Caudrey–Dodd–Gibbon–Sawada–Kotera equation; Hirota bilinear form; N-soliton solutions
Corresponding author: Aly Seadawy, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

[1] K. Hosseini, P. Mayeli, and R. Ansari, “Modified Kudryashov method for solving the conformable time-fractional Klein-Gordon equations with quadratic and cubic nonlinearities,” Optik – Int. J. Light Electron Optic., vol. 130, pp. 737–742, 2017. https://doi.org/10.1016/j.ijleo.2016.10.136.Search in Google Scholar

[2] K. Hosseini, P. Mayeli, and R. Ansari, “Bright and singular soliton solutions of the conformable time-fractional Klein-Gordon equations with different nonlinearities,” Waves Random Complex Media, vol. 28, pp. 426–434, 2018. https://doi.org/10.1080/17455030.2017.1362133.Search in Google Scholar

[3] M. Eslami and M. Mirzazadeh, “Optical solitons with Biswas–Milovic equation for power law and dual-power law nonlinearities,” Nonlinear Dynam., vol. 83, nos 1–2, pp. 731–738, 2016. https://doi.org/10.1007/s11071-015-2361-1.Search in Google Scholar

[4] M. Mirzazadeh, M. Eslami, and A. Biswas, “Soliton solutions of the generalized Klein–Gordon equation by using (G′/G)-expansion method,” Comput. Appl. Math., vol. 33, no. 3, pp. 831–839, 2014. https://doi.org/10.1007/s40314-013-0098-3.Search in Google Scholar

[5] H. F. Ismael, H. Bulut, and H. M. Baskonus, “Optical soliton solutions to the Fokas–Lenells equation via sine-Gordon expansion method and (m + (G′/G))-expansion method,” Pramana - J. Phys., vol. 94, no. 1, 2020. https://doi.org/10.1007/s12043-019-1897-x.Search in Google Scholar

[6] W. Gao, H. F. Ismael, S. A. Mohammed, H. M. Baskonus, and H. Bulut, “Complex and real optical soliton properties of the paraxial nonlinear Schrödinger equation in Kerr media with M-fractional,” Front. Physiol., vol. 7, p. 197, 2019. https://doi.org/10.3389/fphy.2019.00197.Search in Google Scholar

[7] Y. Shang, “Bäcklund transformation, Lax pairs and explicit exact solutions for the shallow water waves equation,” Appl. Math. Comput., vol. 187, no. 2, pp. 1286–1297, 2007. https://doi.org/10.1016/j.amc.2006.09.038.Search in Google Scholar

[8] N. Nasreen, A. R. Seadawy, and D. Lu, “Construction of soliton solutions for modified Kawahara equation arising in shallow water waves using novel techniques,” Int. J. Mod. Phys. B, vol. 34, p. 2050045, 2020. https://doi.org/10.1142/s0217979220500459.Search in Google Scholar

[9] L. Guo, Y. Zhang, S. Xu, Z. Wu, and J. He, “The higher order rogue wave solutions of the Gerdjikov-Ivanov equation,” Phys. Scripta, vol. 89, p. 035501, 2014. https://doi.org/10.1088/0031-8949/89/03/035501.Search in Google Scholar

[10] L. Ling, B. F. Feng, and Z. Zhu, “General soliton solutions to a coupled Fokas–Lenells equation,” Nonlinear Anal. R. World Appl., vol. 582, pp. 327–338, 2018.10.1016/j.nonrwa.2017.08.013Search in Google Scholar

[11] X. R. Hu, Y. Chen, and L. J. Qian, “Full symmetry groups and similar reductions of a (2 + 1)-dimensional resonant Davey–Stewartson system,” Commun. Theor. Phys., vol. 55, p. 737, 2011. https://doi.org/10.1088/0253-6102/55/5/01.Search in Google Scholar

[12] H. H. Abdulkareem, H. F. Ismael, E. S. Panakhov, and H. Bulut, “Some novel solutions of the coupled Whitham–Broer–Kaup equations,” in International Conf. on Computational Mathematics and Engineering Sciences, 2019, pp. 200–208.10.1007/978-3-030-39112-6_14Search in Google Scholar

[13] H. F. Ismael and H. Bulut, “On the solitary wave solutions to the (2 + 1)-dimensional Davey–Stewartson equations,” in International Conf. on Computational Mathematics and Engineering Sciences, 2019, pp. 156–165.10.1007/978-3-030-39112-6_11Search in Google Scholar

[14] M. Alquran and K. Al-Khaled, “Mathematical methods for a reliable treatment of the (2 + 1)-dimensional Zoomeron equation,” Math. Sci., vol. 6, p. 11, 2013. https://doi.org/10.1186/2251-7456-6-11.Search in Google Scholar

[15] N. Cheemaa, S. Chen, and A. R. Seadawy, “Propagation of isolated waves of coupled nonlinear (2 + 1)-dimensional Maccari system in plasma physics,” Results Phys., vol. 17, p. 102987, 2020. https://doi.org/10.1016/j.rinp.2020.102987.Search in Google Scholar

[16] W. Gao, H. F. Ismael, H. Bulut, and H. M. Baskonus, “Instability modulation for the (2 + 1)-dimension paraxial wave equation and its new optical soliton solutions in Kerr media,” Phys. Scripta, vol. 95, p. 035207, 2019. https://doi.org/10.1088/1402-4896/ab4a50.Search in Google Scholar

[17] E. V. Krishnan, A. Biswas, Q. Zhou, and M. Alfiras, “Optical soliton perturbation with Fokas–Lenells equation by mapping methods,” Optik, vol. 178, p. 104, 2019. https://doi.org/10.1016/j.ijleo.2018.10.017.Search in Google Scholar

[18] V. O. Vakhnenko, E. J. Parkes, and A. J. Morrison, “A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation,” Chaos, Solit. Fractals, vol. 17, p. 683, 2003. https://doi.org/10.1016/s0960-0779(02)00483-6.Search in Google Scholar

[19] M. Arshad, A. R. Seadawy, D. Lu, and J. Wang, “Travelling wave solutions of Drinfel’d–Sokolov–Wilson, Whitham–Broer–Kaup and (2 + 1)-dimensional Broer–Kaup–Kupershmit equations and their applications,” Chin. J. Phys., vol. 55, no. 3, pp. 780–797, 2017. https://doi.org/10.1016/j.cjph.2017.02.008.Search in Google Scholar

[20] M. Arshad, A. R. Seadawy, and D. Lu, “Elliptic function and solitary wave solutions of the higher-order nonlinear Schrödinger dynamical equation with fourth-order dispersion and cubic-quintic nonlinearity and its stability,” Eur. Phys. J. Plus, vol. 132, p. 1, 2017. https://doi.org/10.1140/epjp/i2017-11655-9.Search in Google Scholar

[21] A. R. Seadawy, M. Arshad, and D. Lu, “Stability analysis of new exact traveling-wave solutions of new coupled KdV and new coupled Zakharov–Kuznetsov systems,” Eur. Phys. J. Plus, vol. 132, p. 162, 2017. https://doi.org/10.1140/epjp/i2017-11437-5.Search in Google Scholar

[22] D. Lu, A. R. Seadawy, and A. Ali, “Applications of exact traveling wave solutions of modified Liouville and the symmetric regularized long wave equations via two new techniques,” Results Phys., vol. 9, pp. 1403–1410, 2018. https://doi.org/10.1016/j.rinp.2018.04.039.Search in Google Scholar

[23] M. A. Helal, A. R. Seadawy, and M. H. Zekry, “Stability analysis of solitary wave solutions for the fourth-order nonlinear Boussinesq water wave equation,” Appl. Math. Comput., vol. 232, pp. 1094–1103, 2014. https://doi.org/10.1016/j.amc.2014.01.066.Search in Google Scholar

[24] R. Hirota and M. Ito, “A direct approach to multi-periodic wave solutions to nonlinear evolution equations,” J. Phys. Soc. Japan, vol. 50, p. 338, 1981. https://doi.org/10.1143/jpsj.50.338.Search in Google Scholar

[25] E. Aksoy, A. C. Çevikel, and A. Bekir, “Soliton solutions of (2 + 1)-dimensional time-fractional Zoomeron equation,” Optik, vol. 127, p. 6933, 2016. https://doi.org/10.1063/1.4952086.Search in Google Scholar

[26] A. Bueno-Orovio, V. M. Pérez-García, and F. H. Fenton, “Spectral methods for partial differential equations in irregular domains: the spectral smoothed boundary method,” SIAM J. Sci. Comput., vol. 28, no. 3, pp. 886–900, 2006. https://doi.org/10.1137/040607575.Search in Google Scholar

[27] A. R. Seadawy and D. Lu, “Bright and dark solitary wave soliton solutions for the generalized higher order nonlinear Schrödinger equation and its stability,” Results Phys., vol. 7, p. 43, 2017. https://doi.org/10.1016/j.rinp.2016.11.038.Search in Google Scholar

[28] J. Manafian and M. F. Aghdaei, “Abundant soliton solutions for the coupled Schrödinger–Boussinesq system via an analytical method,” Eur. Phys. J. Plus, vol. 131, p. 97, 2016. https://doi.org/10.1140/epjp/i2016-16097-3.Search in Google Scholar

[29] A. Biswas, M. Ekici, A. Sonmezoglu, et al.., “Optical solitons with Lakshmanan–Porsezian–Daniel model using a couple of integration schemes,” Optik, vol. 158, p. 705, 2018.10.1016/j.ijleo.2017.12.190Search in Google Scholar

[30] H. M. Baskonus, T. A. Sulaiman, and H. Bulut, “On the novel wave behaviors to the coupled nonlinear Maccari’s system with complex structure,” Optik, vol. 131, p. 1036, 2017. https://doi.org/10.1016/j.ijleo.2016.10.135.Search in Google Scholar

[31] K. K. Ali, A. R. Seadawy, A. Yokusy, R. Yilmazer, and H. Bulut, “Propagation of dispersive wave solutions for (3 + 1)-dimensional nonlinear modified ZakharovKuznetsov equation, plasma physics,” Int. J. Mod. Phys. B, vol. 34, p. 2050227, 2020. https://doi.org/10.1142/s0217979220502276.Search in Google Scholar

[32] K. K. Ali, R. Yilmazer, and H. Bulut, “Analytical solutions to the coupled Boussinesq–Burgers equations via sine-Gordon expansion method,” in 4th International Conf. on Computational Mathematics and Engineering Sciences (CMES-2019), p. 233.10.1007/978-3-030-39112-6_17Search in Google Scholar

[33] M. Shakeel and S. T. Mohyud-Din, “Solution of fifth order Caudrey–Dodd–Gibbon–Sawada–Kotera equation by the alternative (G′/G)-expansion method with generalized Riccati equation,” Walailak J. Sci. Technol., vol. 12, no. 10, pp. 949–960, 2015.Search in Google Scholar

[34] X. Geng, G. He, and L. Wu, “Riemann theta function solutions of the Caudrey–Dodd–Gibbon–Sawada–Kotera hierarchy,” J. Geom. Phys., vol. 140, pp. 85–103, 2019. https://doi.org/10.1016/j.geomphys.2019.01.005.Search in Google Scholar

[35] Q. M. Al-Mdallal and M. I. Syam, “Sine–cosine method for finding the soliton solutions of the generalized fifth-order nonlinear equation,” Chaos, Solit. Fractals, vol. 33, no. 5, pp. 1610–1617, 2007. https://doi.org/10.1016/j.chaos.2006.03.039.Search in Google Scholar

[36] S. Saha Ray, “New soliton solutions of conformable time fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation in modeling wave phenomena,” Mod. Phys. Lett. B, vol. 33, no. 18, p. 1950202, 2019. https://doi.org/10.1142/s0217984919502026.Search in Google Scholar

[37] Y. Li-Jun and L. Ji, “Different-periodic travelling wave solutions for nonlinear equations,” Commun. Theor. Phys., vol. 41, no. 4, p. 481, 2004. https://doi.org/10.1088/0253-6102/41/4/481.Search in Google Scholar

[38] Y.-G. Xu, X.-W. Zhou, and L. Yao, “Solving the fifth order Caudrey–Dodd–Gibbon (CDG) equation using the exp-function method,” Appl. Math. Comput., vol. 206, no. 1, pp. 70–73, 2008. https://doi.org/10.1016/j.amc.2008.08.052.Search in Google Scholar

[39] Q.-X. Qu, B. Tian, K. Sun, and Y. Jiang, “Bäcklund transformation, Lax pair, and solutions for the Caudrey–Dodd–Gibbon equation,” J. Math. Phys., vol. 52, no. 1, p. 13511, 2011. https://doi.org/10.1063/1.3532766.Search in Google Scholar

[40] X. Cheng, Y. Yang, B. Ren, and J. Wang, “Interaction behavior between solitons and (2+ 1)-dimensional CDGKS waves,” Wave Motion, vol. 86, pp. 150–161, 2019. https://doi.org/10.1016/j.wavemoti.2018.08.008.Search in Google Scholar

[41] R. N. Aiyer, B. Fuchssteiner, and W. Oevel, “Solitons and discrete eigenfunctions of the recursion operator of non-linear evolution equations. I. The Caudrey–Dodd–Gibbon–Sawada–Kotera equation,” J. Phys. A Math. Gen., vol. 19, no. 18, p. 3755, 1986. https://doi.org/10.1088/0305-4470/19/18/022.Search in Google Scholar

[42] D. Lu, A. R. Seadawy, and A. Ali, “Dispersive traveling wave solutions of the equal-width and modified equal-width equations via mathematical methods and its applications,” Results Phys., vol. 9, pp. 313–320, 2018. https://doi.org/10.1016/j.rinp.2018.02.036.Search in Google Scholar

[43] M. Arshad, A. Seadawy, and D. Lu, “Modulation stability and optical soliton solutions of nonlinear Schrodinger equation with higher order dispersion and nonlinear terms and its applications,” Superlattice. Microst., vol. 112, pp. 422–434, 2017. https://doi.org/10.1016/j.spmi.2017.09.054.Search in Google Scholar

[44] A. Ali, A. R. Seadawy, and D. Lu, “Computational methods and traveling wave solutions for the fourth-order nonlinear Ablowitz–Kaup–Newell–Segur water wave dynamical equation via two methods and its applications,” Open Phys. J., vol. 16, pp. 219–226, 2018. https://doi.org/10.1515/phys-2018-0032.Search in Google Scholar

[45] M. Iqbal, A. R. Seadawy, O. H. Khalil, and D. Lu, “Propagation of long internal waves in density stratified ocean for the (2 + 1)-dimensional nonlinear Nizhnik–Novikov–Vesselov dynamical equation,” Results Phys., vol. 16, p. 102838, 2020. https://doi.org/10.1016/j.rinp.2019.102838.Search in Google Scholar

[46] Y. G. Ozkan, E. Yasar, and A. Seadawy, “A third-order nonlinear Schrodinger equation: the exact solutions, group-invariant solutions and conservation laws,” J. Taibah Univ. Sci., vol. 14, no. 1, pp. 585–597, 2020. https://doi.org/10.1080/16583655.2020.1760513.Search in Google Scholar

[47] N. Farah, A. R. Seadawy, S. Ahmad, S. T. Raza Rizvi, and M. Younis, “Interaction properties of soliton molecules and Painleve analysis for nano bioelectronics transmission model,” Opt. Quant. Electron., vol. 52, 2020, Art no. 329, pages 1–15 https://doi.org/10.1007/s11082-020-02443-0.Search in Google Scholar

[48] S. T. Raza Rizvi, A. R. Seadawy, I. Ali, I. Bibi, and M. Younis, “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, vol. 34, no. 35, 2020, Art no. 2050399, pages 15. https://doi.org/10.1142/s0217984920503996.Search in Google Scholar

[49] I. Ali, A. R. Seadawy, S. T. Raza Rizvi, M. Younis, and K. Ali, “Conserved quantities along with Painleve analysis and optical solitons for the nonlinear dynamics of Heisenberg ferromagnetic spin chains model,” Int. J. Mod. Phys. B, vol. 34, no. 30, 2020, Art no. 2050283, pages 15. https://doi.org/10.1142/s0217979220502835.Search in Google Scholar
Received: 2020-07-24
Revised: 2021-01-25
Accepted: 2021-02-07
Published Online: 2021-03-11
Published in Print: 2023-02-23

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/ijnsns-2020-0169
Keywords for this article
breather solutions; Caudrey–Dodd–Gibbon–Sawada–Kotera equation; Hirota bilinear form; N-soliton solutions

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Modeling and assessment of the flow and air pollutants dispersion during chemical reactions from power plant activities
  4. Stochastic dynamics of dielectric elastomer balloon with viscoelasticity under pressure disturbance
  5. Unsteady MHD natural convection flow of a nanofluid inside an inclined square cavity containing a heated circular obstacle
  6. Fractional-order generalized Legendre wavelets and their applications to fractional Riccati differential equations
  7. Battery discharging model on fractal time sets
  8. Adaptive neural network control of second-order underactuated systems with prescribed performance constraints
  9. Optimal control for dengue eradication program under the media awareness effect
  10. Shifted Legendre spectral collocation technique for solving stochastic Volterra–Fredholm integral equations
  11. Modeling and simulations of a Zika virus as a mosquito-borne transmitted disease with environmental fluctuations
  12. Mathematical analysis of the impact of vaccination and poor sanitation on the dynamics of poliomyelitis
  13. Anti-sway method for reducing vibrations on a tower crane structure
  14. Stable soliton solutions to the time fractional evolution equations in mathematical physics via the new generalized G ′ / G -expansion method
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  16. An estimative (warning) model for recognition of pandemic nature of virus infections
  17. Interaction among a lump, periodic waves, and kink solutions to the KP-BBM equation
  18. Global exponential stability of periodic solution of delayed discontinuous Cohen–Grossberg neural networks and its applications
  19. An efficient class of fourth-order derivative-free method for multiple-roots
  20. Numerical modeling of thermal influence to pollutant dispersion and dynamics of particles motion with various sizes in idealized street canyon
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  22. Delay-dependent robust stability analysis of uncertain fractional-order neutral systems with distributed delays and nonlinear perturbations subject to input saturation
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  25. Infinite line of equilibriums in a novel fractional map with coexisting infinitely many attractors and initial offset boosting
  26. Lump solutions to a generalized nonlinear PDE with four fourth-order terms
  27. Quantum motion control for packaging machines
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