THERMAL SCIENCE

International Scientific Journal

Mukhtar Y. Y. Abdalla

,

Bakery A.e Younies

,

Haroon D.S. Adam

,

Ahmed M.i Adam

,

Yagoub A.s Arko

,

Mohammad S Jazmati

MULTIPLE SOLITON SOLUTIONS OF THE MODEL MKDV WITH TIME-DEPENDENT VARIABLE COEFFICIENTS

ABSTRACT
We investigate exact multiple soliton waves in the time-dependent variable coefficients modified KdV (mKdV) equation. Employing similarity transformations, as well as tanh and sech function methods, we present multiple kink wave, singular kink, and triangular function solutions for the time-dependent mKdV equation with variable coefficients. Specifically. We investigate the construction of multiple solitons by choosing a special form of the time variable. Our results demonstrate that the specific choice of these analogous temporal variables can effectively control the characteristics of numerous soliton kinks, offering potential applications in diverse fields
KEYWORDS
mKdV equations, exact solutions, similarity transformation
PAPER SUBMITTED: 2024-06-20
PAPER REVISED: 2024-10-10
PAPER ACCEPTED: 2024-10-30
PUBLISHED ONLINE: 2025-01-25
DOI REFERENCE: https://doi.org/10.2298/TSCI2406029A
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2024, VOLUME 28, ISSUE No. 6, PAGES [5029 - 5036]
REFERENCES
[1] Khoury, S. A., Soliton and Periodic Solutions for Higher Order Wave Equations of KdV Type (I), Chaos Solitons Fractals, 26 (2005), 1, pp. 25-32, 10.1016/j.chaos.2004.12.027
[2] Sun, B., Wazwaz, A. M., General High-Order Breathers and Rogue Waves in the (3+1)-Dimensional KP-Boussinesq Equation, Commun. Non-Linear Sci. Numer. Simul., 64 (2018), Nov., pp. 1-13, 10.1016/j.cnsns.2018.04.005
[3] Huang, C., et al., Stabilization of Multipole-Mode Solitons in Mixed Linear-Non-Linear Lattices with a PT Symmetry, Opt. Express, 21 (2013), 3, pp. 3917-3925, 10.1364/oe.21.003917
[4] Biswas, A., Conservation Laws for Optical Solitons with Anti-Cubic and Generalized Anti-Cubic Non-Linearities, Optik, 176 (2019), Jan., pp. 198-201, 10.1016/j.ijleo.2018.09.074
[5] Wazwaz, A. M., New (3+1)-Dimensional Equation with KdV Equation Constituting Its Main Part: Multiple Soliton Solutions, Math. Methods Appl. Sci., 39 (2016), 4, pp. 886-891, 10.1002/mma.3528
[6] Hirota, R., The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, UK, 2004, 10.1017/cbo9780511543043
[7] Hereman, W., Nuseir, A., Symbolic Methods to Construct Exact Solutions of Non-Linear Partial Differential Equations, Math. Comput. Simul., 43 (1997), 1, pp. 13-27, 10.1016/s0378-4754(96)00053-5
[8] Verheest, F., et al., Modified Korteweg-de Vries Solitons at Supercritical Densities in Two-Electron Temperature Plasmas, J. Plasma Phys., 82 (2016), 905820208, 10.1017/s0022377816000349
[9] Wazwaz, A. M., Partial Differential Equations and Solitary Waves Theory, Springer and HEP, Berlin and Peking, Germany and China, 2009, 10.1007/978-3-642-00251-9
[10] Rotschild, C., et al., The 2-D Multipole Solitons in Non-Local Non-Linear Media, Opt. Lett., 31 (2006) 22, pp. 3312-3314, 10.1364/ol.31.003312
[11] Gardner, C. S., et al., Korteweg-de Vries Equation and Generalizations, VI Methods for Exact Solution, Commun.Pure Appl. Math. 27 (1974), 1, pp 97-133, 10.1002/cpa.3160270108
[12] Abdel-Rahman, A. M. M., Simple Multisoliton Solutions, Am. J. Phys., 51 (1983), 510, 10.1119/1.13218
[13] Zhang, J., Simple Soliton Solution Method for the Combined KdV and MKdV Equation, Int. J. Theor. Phys., 39 (2000), June, pp. 1697-1702
[14] Hassanien, I. A., et al., Multicnoidal and Multitravelling Wave Solutions for Some Non-Linear Equations of Mathematical Physics, Physica Scripta, 67 (2003), 6, pp 457-463, 10.1238/physica.regular.067a00457
[15] Wazwaz, A. M., Multiple Real and Multiple Complex Soliton Solutions for the Integrable Sine-Gordon Equation, Optik, 172 (2018), Nov., pp. 622-627, 10.1016/j.ijleo.2018.07.080
[16] Wazwaz, A. M., THe N-Soliton Solutions for the Sine-Gordon Equation of Different Dimensions, J. Appl. Math. Inform., 30 (2012), 5-6, pp. 925-934
[17] Wazwaz, A. M., One and Two Soliton Solutions for The Sinh-Gordon Equation in (1+1), (2+1), and (3+1) Dimensions, Appl. Math. Lett., 25 (2012), 12, pp. 2354-2358, 10.1016/j.aml.2012.07.001
[18] Yang, Z., Zhong, W. P., Analytical Solutions To Sine-Gordon Equation With Variable Coefficient, Roman. Rep. Phys. 66 (2014), 2, pp. 262-273
[19] Biswas, A., Optical Soliton Perturbation with Radhakrishnan-Kundu-Lakshmanan Equation by Traveling Wave Hypothesis, Optik, 171 (2018), Oct., pp. 217-220, 10.1016/j.ijleo.2018.06.043
[20] Biswas, A., Chirp-Free Bright Optical Solitons and Conservation Laws for Complex Ginzburg-Landau Equation with Three Non-Linear Forms, Optik, 174 (2018), Dec., pp. 207-215, 10.1016/j.ijleo.2018.08.063
[21] Yao, Y., et al., The New Integrable Deformations of a Short Pulse Equation and Sine-Gordon Equation, and Their Solutions, J. Phys. A: Math. Theor., 44 (2011), 065201, 10.1088/1751-8113/44/6/065201
[22] Xin, J. X., Modelling Light Bullets with the 2-D Sine-Gordon Equation, Physica D, 135 (2000), 3-4, pp. 345-368, 10.1016/s0167-2789(99)00128-1
[23] Kaur, L., Wazwaz, A. M., Optical Solitons for Pertuber Gerdjekov-Ivanov Equation, Optik, 174 (2018), Dec., pp. 447-451, 10.1016/j.ijleo.2018.08.072
[24] Malfliet, W., Simple Solution Method for Pertuber Korteweg-de Vries Equation, Am. J. Phys., 49 (1981), 7, pp. 666- 668, 10.1119/1.12422
PDF VERSION [DOWNLOAD]
Multiple soliton solutions of the model mKdV with time-dependent variable coefficients

© 2026 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence