New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation with Higher Order Dispersion

Asit Saha; Seydi Battal Gazi Karakoç; Khalid K. Ali

doi:10.36753/mathenot.1180832

EN

New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation with Higher Order Dispersion

Abstract

The aim of the present paper is to obtain and analyze new exact travelling wave solutions and bifurcation behavior of modified Zakharov-Kuznetsov (mZK) equation with higher-order dispersion term. For this purpose, the first and second simplest methods are used to build soliton solutions of travelling wave solutions. Furthermore, the bifurcation behavior of traveling waves including new types of quasiperiodic and multi-periodic traveling wave motions have been examined depending on the physical parameters. Multistability for the nonlinear mZK equation has been investigated depending on fixed values of physical parameters with various initial conditions. The suggested methods for the analytical solutions are powerful and beneficial tools to obtain the exact travelling wave solutions of nonlinear evolution equations (NLEEs). Two and three-dimensional plots are also provided to illustrate the new solutions. Bifurcation and multistability behaviors of traveling wave solution of the nonlinear mZK equation with higher-order dispersion will add some value to the literature of mathematical and plasma physics.

Keywords

Bifurcation, First simplest method, Modified Zakharov-Kuznetsov equation, Second simplest method, Quasiperiodic motion

References

[1] Jhangeer, A., Hussain, A., Tahir, S., Sharif, S.: Solitonic, super nonlinear, periodic, quasiperiodic, chaotic waves and conservation laws of modified Zakharov-Kuznetsov equation in transmission line. Commun. Nonlinear Sci. Numer. Simul. 86, (2020).
[2] Khalfallah, M.: New Exact traveling wave solutions of the (2+1) dimensional Zakharov-Kuznetsov (ZK) equation. An. St. Univ. Ovidius Constanta. 15(2), 35–44 (2007).
[3] Ali, M. N., Seadawy, A. R., Husnine, S. M.: Lie point symmetries, conservation laws and exact solutions of (1 + n)-dimensional modified Zakharov–Kuznetsov equation describing the waves in plasma physics. Pramana - J. Phys. 91(48), 1-9 (2018).
[4] Batiha, K.: Approximate analytical solution for the Zakharov–Kuznetsov equations with fully nonlinear dispersion. Journal of Computational and Applied Mathematics. 216, 157-163 (2008).
[5] Ablowitz, M. X., Clarkson, P. A.: Solitons, nonlinear evolution equations and inverse scattering transform. Cambridge: Cambridge University Press, (1990).
[6] Vakhnenko, V. O., Parkes, E. J., Morrison, A. J.: A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation. Chaos Solitons and Fractals. 17(4), 683-692 (2003).
[7] Rogers, C., Shadwick, W. F.: Bäcklund Transformations and Their Applications, Mathematics in Science and Engineering. Academic Press, New York, NY, USA (1982).
[8] Jawad, A. J. M., Petkovic, M. D., Biswas, A.: Modified simple equation method for nonlinear evolution equations. Applied Mathematics and Computation. 217, 869–877 (2010).
[9] Wang, M. L., Zhou, Y. B., Li, Z. B.: Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Physics Letters Section A. 216(1–5), 67–75 (1996).
[10] Seadawy, A. R., El-Rashidy, K.: Travelling wave solutions for some coupled nonlinear evolution equations by using the direct algebraic method. Mathematical and Computer Modelling. 57, 1371–1379 (2013).

[11] Khater, M. M. A.: A hybrid analytical and numerical analysis of ultra-short pulse phase shifts. Chaos, Solitons and Fractals. 169, 113232 (2023).
[12] Hirota, R.: Exact solution of the korteweg-de vries equation for multiple collisions of solitons. Physical Review Letters. 27(18), 1192–1194 (1971).
[13] Malfliet,W., Hereman,W.: The tanh method. I: Exact solutions of nonlinear evolution and wave equations. Physica Scripta. 54(6), 563-568 (1996).
[14] Wazwaz, A. M.: The tanh method for travelling wave solutions of nonlinear equations. Applied Mathematics and Computation. 154(3), 713-723 (2004).
[15] El-Wakil, S. S., Abdou, M. A.: New exact travelling wave solutions using modified extended tanh-function method. Chaos Solitons and Fractals. 31(4), 840-852 (2007).
[16] Fan, E.: Extended tanh-function method and its applications to nonlinear equations. Physics Letters A. 277(4-5), 212-218 (2000).
[17] Wazwaz, A. M.: The extended tanh method for abundant solitary wave solutions of nonlinear wave equations. Applied Mathematics and Computation. 187(2), 1131-1142 (2007).
[18] Liu, S., Fu, Z., Liu, S., Zhao, Q.: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Physics Letters A. 289(1-2), 69–74 (2001).
[19] Al-Muhiameed, A. Z. I., Abdel-Salam, E. A. B.: Generalized Jacobi elliptic function solution to a class of nonlinear Schrödinger-type equations. Mathematical Problems in Engineering. 2011 11 pages (2011).
[20] Gepreel K. A., Shehata, A. R.: Jacobi elliptic solutions for nonlinear differential difference equations in mathematical physics. Journal of Applied Mathematics. 2012 15 pages (2012).
[21] Yan Z., Zhang, H.: New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water. Physics Letters A. 285(5-6), 355–362 (2001).
[22] Wazwaz, A. M.: A sine–cosine method for handling nonlinear wave equations. Mathematical and Computer Modelling. 40(5-6), 499-508(2004).
[23] Wang, M., Li, X.: Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation. Chaos, Solitons and Fractals. 24(5), 1257–1268 (2005).
[24] Abdou, M. A.: The extended F-expansion method and its application for a class of nonlinear evolution equations. Chaos, Solitons and Fractals. 31(1), 95–104 (2007).
[25] Khater, M. M. A.: Nonlinear elastic circular rod with lateral inertia and finite radius: Dynamical attributive of longitudinal oscillation. International Journal of Modern Physics B. 37(6) 2350052 (2023).
[26] Fan, E., Zhang, H.: A note on the homogeneous balance method. Physics Letters A. 246(5), 403-406 (1998).
[27] He, J. H.,Wu, X. H.: Exp-function method for nonlinear wave equations. Chaos, Solitons and Fractals. 30(3), 700–708 (2006).
[28] Naher, H., Abdullah, F. A., Akbar, M. A.: New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the exp-function method. Journal of Applied Mathematics. 2012 (2012)14 pages.
[29] Mohyud-Din, S. T., Noor, M. A.: Exp-function method for traveling wave solutions of modified Zakharov-Kuznetsov equation. Journal of King Saud University. 22(4), 213–216(2010).
[30] Esen, A., Yagmurlu, N. M., Tasbozan, O.: Double exp-function method for multisoliton solutions of the Tzitzeica-Dodd-Bullough equation, Acta Mathematicae Applicatae Sinica, English Series. 32(2), 461-468 (2016).
[31] Salas, A. H., Gomez, C. A.: Application of the Cole-Hopf transformation for finding exact solutions to several forms of the seventh-order KdV equation. Mathematical Problems in Engineering. 2010 (2010) 14 pages.
[32] Ugurlu, Y., Kaya, D., Inan, I. E.: Comparison of three semianalytical methods for solving (1 + 1)-dimensional dispersive long wave equations. Computers & Mathematics with Applications. 61(5), 1278–1290 (2011).
[33] Dinarvand, S., Khosravi, S., Doosthoseini, A., Rashidi, M. M.: The homotopy analysis method for solving the Sawada-Kotera and Laxs fifth-order KdV equations. Advances in Theoretical and Applied Mechanics. 1, 327–335 (2008).
[34] Esen, A., Tasbozan, O., Yagmurlu, N. M.: Approximate Analytical Solutions of the Fractional Sharma-Tasso-Olver Equation Using Homotopy Analysis Method and a Comparison with Other Methods, Cankaya University Journal of Science and Engineering. 9(2), 139-147 (2012).
[35] Biazar, J., Badpeima, F., Azimi, F.: Application of the homotopy perturbation method to Zakharov-Kuznetsov equations. Computers Mathematics with Applications. 58, 2391–2394 (2009).
[36] Zafar, A., Raheel, M., Jafar, N., Nisar, K. S.: Soliton solutions to the DNA Peyrard Bishop equation with beta-derivative via three distinctive approache. The European Physical Journal Plus. 135, (2020).
[37] Ali, K. K., Wazwaz, A. M., Mehanna, M. S., Osman, M. S.: On short-range pulse propagation described by (2 +1)-dimensional Schrodinger’s hyperbolic equation in nonlinear optical fibers,. Physica Scripta. 95(7), 075203 (2020).
[38] Wu, X., Rui, W., Hong, X.: Exact travelling wave solutions of explicit type, implicit type and parametric type for K(m,n)equation, Journal of Applied Mathematics. 2012, 23 pages (2012) .
[39] Zhang, R.: Bifurcation analysis for a kind of nonlinear finance system with delayed feedback and its application to control of chaos. Journal of Applied Mathematics. 2012, 18 pages (2012) .
[40] Tascan, F., Bekir, A., Koparan, M.: Travelling wave solutions of nonlinear evolution equations by using the first integral method. Communications in Nonlinear Science and Numerical Simulation. 14, 1810–1815 (2009).
[41] Munro, S., Parkes, E. J.: The derivation of a modified Zakharov-Kuznetsov equation and the stability of its solutions. Journal of Plasma Physics. 62, 305-317 (1999).
[42] Munro, S., Parkes, E. J.: Stability of solitary-wave solutions to a modified Zakharov-Kuznetsov equation. Journal of Plasma Physics. 64, 411-426 (2000).
[43] Khater, M. M. A.: In solid physics equations, accurate and novel soliton wave structures for heating a single crystal of sodium fluoride.International Journal of Modern Physics B. 377 2350068 (2023).
[44] Zakharov, V. E., Kuznetsov, E. A.: On three-dimensional solitons. Soviet Physics. 39, 285288 (1974).
[45] Hesam, S., Nazemi, A., Haghbin, A.: Analytical solution for the Zakharov-Kuznetsov equations by differential transform method,. International Journal of Mathematical and Computational Sciences. 5(3), 496-501(2011).
[46] Schamel, H.: A modified Korteweg–de Vries equation for ion acoustic waves due to resonant electrons. Journal of Plasma Physics. 9, 377–387 (1973).
[47] Liu, Y., Teng, Q., Tai, W., Zhou, J., Wang, Z.: Symmetry reductions of the (3 + 1)-dimensional modified Zakharov–Kuznetsov equation. Advances in Difference Equations. 2019, 2-14 (2019).
[48] Park, C., Khater, M. M. A., Abdel-Aty, A. H., Attia, R. A. M., Lu, D. M.: On new computational and numerical solutions of the modified Zakharov–Kuznetsov equation arising in electrical engineering. Alexandria Engineering Journal. 59, 1099–1105 (2020).
[49] Peng, Y. Z.: Exact Travelling Wave Solutions for a Modified Zakharov-Kuznetsov Equation. Acta Physica Polonica A. 115(3), 609-612 (2009).
[50] Natiq, H., Said, M. R. M., Ariffin, M. R. K., He, S., Rondoni, L., Banerjee, S.: Self-excited and hidden attractors in a novel chaotic system with complicated multistability. The European Physical Journal Plus. 133, 557 (2018).
[51] He, S., Banerjee, S., Sun, K.: Complex dynamics and multiple coexisting attractors in a fractional-order microscopic chemical system,. The European Physical Journal Special Topics. 228, 195-207 (2019).
[52] Arecchi, F. T., Meucci, R., Puccioni, G., Tredicce, J.: Experimental Evidence of Subharmonic Bifurcations, Multistability, and Turbulence in a Q-Switched Gas Laser. Physical Review Letters. 49, 1217 (1982).
[53] Natiq, H., Banerjee, S., Misra, A. P., Said, M. R. M.: Degenerating the butterfly attractor in a plasma perturbation model using nonlinear controllers. Chaos Solitons and Fractals. 122, 58-68 (2019).
[54] Morfu, S., Nofiele, B., Marquie, P.: On the use of multistability for image processing. Physics Letters A. 367, 192-198 (2007).
[55] Rahim, M. F. A., Natiq, H., Fataf, N. A. A., Banerjee, S.: Dynamics of a new hyperchaotic system and multistability. The European Physical Journal Plus. 134, 499 (2019).
[56] Li, C., Sprott, J. C.: Multistability in the Lorenz System: A Broken Butterfly. International Journal of Bifurcation and Chaos. 24, 1450131 (2014).
[57] Yong, J., Haida, W., Changxuan, Y.: Multistability Phenomena in Discharge Plasma. Chinese Physics Letters. 5, 201-204 (1988).
[58] Hahn, S. J., Pae, K. H.: Competing multistability in a plasma diode. Physics of Plasmas. 10, 314 (2003).
[59] Prasad, P. K., Gowrishankar, A., Saha, A., Banerjee, S.: Dynamical properties and fractal patterns of nonlinear waves in solar wind plasma. Physica Scripta. 6(95), (2020).
[60] Abdikian, A., Tamang, J., Saha, A.: Electron-acoustic supernonlinear waves and their multistability in the framework of the nonlinear Schrödinger equation. Communication in Theoretical Physics. 72, 075502(2020).
[61] Pradhan, B., Saha, A., Natiq, H., Banerjee, S.: Multistability and chaotic scenario in a quantum pair-ion plasma. Zeitschrift für Naturforschung A. 76(2), 109-119 (2020).
[62] Saha, A., Pradhan, B., Banerjee, S.: Multistability and dynamical properties of ion-acoustic wave for the nonlinear Schrödinger equation in an electron–ion quantum plasma. Physica Scripta. 5(95), 055602(2020).
[63] Kudryashov, N. A.: Highly dispersive solitary wave solutions of perturbed nonlinear Schrodinger equations. Applied Mathematics and Computation. 371, 124972 (2020).
[64] Saha, A., Ali, K. K., Rezazadeh, H., Ghatani, Y.: Analytical optical pulses and bifurcation analysis for the traveling optical pulses of the hyperbolic nonlinear Schrödinger equation. Optical and Quantum Electronics. 53, 150 (2021).
[65] Lakshmanan, M., Rajasekar, S.: Nonlinear Dynamics. Heidelberg, Springer-Verlag, (2003).
[66] Saha, A.: Bifurcation of travelling wave solutions for the generalized KP-MEW equations. Communications in Nonlinear Science and Numerical Simulation. 17, 3539 (2012).
[67] Karakoc, S.B.G., Saha, A., Sucu, D.: A novel implementation of Petrov-Galerkin method to shallow water solitary wave pattern and superperiodic traveling wave and its multistability: generalized Korteweg-de Vries equation. Chinese Journal of Physics. 68, 605-617(2020).
[68] Saha, A.: Bifurcation, periodic and chaotic motions of the modified equal width burgers (MEW-Burgers) equation with external periodic perturbation. Nonlinear Dynamics. 87, 2193-2201(2017).
[69] Saha, A., Banerjee, S.: Dynamical Systems and Nonlinear Waves in Plasmas. CRC Press-Boca Raton, (2021).
[70] Guckenheimer, J., Holmes, P. J.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. New York, Springer-Verlag (1983).

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Asit Saha
0000-0001-7542-7699
India

Seydi Battal Gazi Karakoç ^*
0000-0002-2348-4170
Türkiye

Khalid K. Ali
0000-0003-3381-0053
Egypt

Early Pub Date

August 6, 2023

Publication Date

October 25, 2023

Submission Date

September 27, 2022

Acceptance Date

April 27, 2023

Published in Issue

Year 2023 Volume: 11 Number: 4

DOI

https://doi.org/10.36753/mathenot.1180832

IZ

https://izlik.org/JA83NE86SL

APA

Saha, A., Karakoç, S. B. G., & Ali, K. K. (2023). New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation with Higher Order Dispersion. Mathematical Sciences and Applications E-Notes, 11(4), 226-240. https://doi.org/10.36753/mathenot.1180832

AMA

1.Saha A, Karakoç SBG, Ali KK. New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation with Higher Order Dispersion. Math. Sci. Appl. E-Notes. 2023;11(4):226-240. doi:10.36753/mathenot.1180832

Chicago

Saha, Asit, Seydi Battal Gazi Karakoç, and Khalid K. Ali. 2023. “New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation With Higher Order Dispersion”. Mathematical Sciences and Applications E-Notes 11 (4): 226-40. https://doi.org/10.36753/mathenot.1180832.

EndNote

Saha A, Karakoç SBG, Ali KK (October 1, 2023) New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation with Higher Order Dispersion. Mathematical Sciences and Applications E-Notes 11 4 226–240.

IEEE

[1]A. Saha, S. B. G. Karakoç, and K. K. Ali, “New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation with Higher Order Dispersion”, Math. Sci. Appl. E-Notes, vol. 11, no. 4, pp. 226–240, Oct. 2023, doi: 10.36753/mathenot.1180832.

ISNAD

Saha, Asit - Karakoç, Seydi Battal Gazi - Ali, Khalid K. “New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation With Higher Order Dispersion”. Mathematical Sciences and Applications E-Notes 11/4 (October 1, 2023): 226-240. https://doi.org/10.36753/mathenot.1180832.

JAMA

1.Saha A, Karakoç SBG, Ali KK. New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation with Higher Order Dispersion. Math. Sci. Appl. E-Notes. 2023;11:226–240.

MLA

Saha, Asit, et al. “New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation With Higher Order Dispersion”. Mathematical Sciences and Applications E-Notes, vol. 11, no. 4, Oct. 2023, pp. 226-40, doi:10.36753/mathenot.1180832.

Vancouver

1.Asit Saha, Seydi Battal Gazi Karakoç, Khalid K. Ali. New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation with Higher Order Dispersion. Math. Sci. Appl. E-Notes. 2023 Oct. 1;11(4):226-40. doi:10.36753/mathenot.1180832

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New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation with Higher Order Dispersion

Abstract

New Exact Soliton Solutions and Multistability for the Modified Zakharov-Kuznetsov Equation with Higher Order Dispersion

Abstract

Keywords

References

Details

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Subjects

Journal Section

Authors

Early Pub Date

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

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Cited By

Bifurcation, chaotic behavior, and traveling wave solution of stochastic coupled Konno–Oono equation with multiplicative noise in the Stratonovich sense

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Superconvergence analysis of a low order expanded conforming mixed finite element method for the extended Fisher–Kolmogorov equation

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