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<p>This study focuses on understanding wave propagation in dispersive and nonlinear media by analyzing the dynamic behavior of the modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation. Building on the extensively researched KdV-ZK equation, we examine solitary wave solutions of the (3+1)-dimensional mKdV–ZK equation using the extended modified Sardar sub-equation method. As a result, a diverse array of novel soliton solutions are obtained, including multi-peak solitons, kink and anti-kink waves, dark and bright solitons, and breather waves. These analytical solutions are derived in terms of key physical quantities such as electrostatic field potential, quantum statistical pressure, electric fields, and magnetic fields. Graphical representations are provided to illustrate the distinct dynamical features of the obtained wave solutions, highlighting their relevance to plasma physics, nonlinear optics, and electromagnetic wave propagation. The results confirm that the extended modified Sardar sub-equation method is an efficient and versatile tool for solving higher-dimensional nonlinear partial differential equations, with broad applicability to nonlinear wave systems in applied mathematics and physical sciences.</p>

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Wave Dynamics and Electromagnetic Influences on Propagation in the (3+1)-Dimensional Modified KdV–Zakharov–Kuznetsov Equation: Theoretical Insights and Applications

  • 期刊论文

  • 发表日期：

    2026年2月9日

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  • Khurrem Shehzad,
  • Jun Wang,
  • Muhammad Arshad,
  • M. Ozair Ahmad,
  • Muhammad Attique

摘要

This study focuses on understanding wave propagation in dispersive and nonlinear media by analyzing the dynamic behavior of the modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation. Building on the extensively researched KdV-ZK equation, we examine solitary wave solutions of the (3+1)-dimensional mKdV–ZK equation using the extended modified Sardar sub-equation method. As a result, a diverse array of novel soliton solutions are obtained, including multi-peak solitons, kink and anti-kink waves, dark and bright solitons, and breather waves. These analytical solutions are derived in terms of key physical quantities such as electrostatic field potential, quantum statistical pressure, electric fields, and magnetic fields. Graphical representations are provided to illustrate the distinct dynamical features of the obtained wave solutions, highlighting their relevance to plasma physics, nonlinear optics, and electromagnetic wave propagation. The results confirm that the extended modified Sardar sub-equation method is an efficient and versatile tool for solving higher-dimensional nonlinear partial differential equations, with broad applicability to nonlinear wave systems in applied mathematics and physical sciences.