Explicit travelling wave solutions to the modified Korteweg-de Vries-Zakharov-Kuznetsov and the time-regularized long-wave equations using two efficient integration techniques

Manal Alqhtani; Rabia Gul; Muhammad Abbas; Alina Alb Lupaş; Khaled M. Saad; Manal Alqhtani; Rabia Gul; Muhammad Abbas; Alina Alb Lupaş; Khaled M. Saad

doi:10.3934/math.2025816

AIMS Mathematics

2025, Volume 10, Issue 8: 18268-18294. doi: 10.3934/math.2025816

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Research article

Explicit travelling wave solutions to the modified Korteweg-de Vries-Zakharov-Kuznetsov and the time-regularized long-wave equations using two efficient integration techniques

1.
Department of Mathematics, College of Sciences and Arts, Najran University, Najran, Saudi Arabia
2.
Department of Mathematics, University of Sargodha, 40100 Sargodha, Pakistan
3.
Department of Mathematics and Computer Science, University of Oradea, 410087 Oradea, Romania

Received: 12 May 2025 Revised: 25 June 2025 Accepted: 22 July 2025 Published: 13 August 2025
MSC : 39A12, 9B62, 33B10, 26A48, 26A51

Numerous scientific disciplines such as fluid dynamics, non linear optics and laser physics are the sources of non-linear evolution equations. The modified Korteweg-de Vries-Zakharov-Kuznetsov equation can be applied to analyze the evolution of ion-acoustic perturbations in magnetized plasma made up of two negative ion ingredients that have distinct temperatures. The analysis of shallow water waves gives rise to the time-regularized long-wave equation. In this work, traveling wave solutions to the non-linear evolution equations are obtained by using the modified auxiliary equation method and the simple mapping method. This study demonstrates the presence of traveling wave solutions for the modified Korteweg-de Vries-Zakharov-Kuznetsov equation and the time-regularized long-wave equation. An ordinary differential equation is transformed into a non linear form by applying the traveling wave solutions, which arise from Lie symmetry with infinite dimensions. The results show how rich the analyzed models are in explicit solutions. This leads to the discovery of exact traveling wave solutions for the proposed study including periodic and quasi-periodic solitons, bright and dark solitons, kink, anti-peakon, bell shape, anti-bell shape, W-shape, and M-shape soliton solutions by using the proposed methods. Adding the solutions into the original equation allows for an analysis of their accuracy. It shows how the free parameters affect the amplitudes and wave characteristics. The study provides thorough two-dimensional (2D) and three-dimensional (3D) graphical illustrations of the outcomes that improve comprehension of their physical attributes and show how effectively the recommended approaches work to solve challenging non-linear equations. It is crucial to remember that the suggested techniques are capable, reliable, and engaging analytical instruments for resolving non linear partial differential equations.
Citation: Manal Alqhtani, Rabia Gul, Muhammad Abbas, Alina Alb Lupaş, Khaled M. Saad. Explicit travelling wave solutions to the modified Korteweg-de Vries-Zakharov-Kuznetsov and the time-regularized long-wave equations using two efficient integration techniques[J]. AIMS Mathematics, 2025, 10(8): 18268-18294. doi: 10.3934/math.2025816

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Abstract

Numerous scientific disciplines such as fluid dynamics, non linear optics and laser physics are the sources of non-linear evolution equations. The modified Korteweg-de Vries-Zakharov-Kuznetsov equation can be applied to analyze the evolution of ion-acoustic perturbations in magnetized plasma made up of two negative ion ingredients that have distinct temperatures. The analysis of shallow water waves gives rise to the time-regularized long-wave equation. In this work, traveling wave solutions to the non-linear evolution equations are obtained by using the modified auxiliary equation method and the simple mapping method. This study demonstrates the presence of traveling wave solutions for the modified Korteweg-de Vries-Zakharov-Kuznetsov equation and the time-regularized long-wave equation. An ordinary differential equation is transformed into a non linear form by applying the traveling wave solutions, which arise from Lie symmetry with infinite dimensions. The results show how rich the analyzed models are in explicit solutions. This leads to the discovery of exact traveling wave solutions for the proposed study including periodic and quasi-periodic solitons, bright and dark solitons, kink, anti-peakon, bell shape, anti-bell shape, W-shape, and M-shape soliton solutions by using the proposed methods. Adding the solutions into the original equation allows for an analysis of their accuracy. It shows how the free parameters affect the amplitudes and wave characteristics. The study provides thorough two-dimensional (2D) and three-dimensional (3D) graphical illustrations of the outcomes that improve comprehension of their physical attributes and show how effectively the recommended approaches work to solve challenging non-linear equations. It is crucial to remember that the suggested techniques are capable, reliable, and engaging analytical instruments for resolving non linear partial differential equations.

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