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<p>The modified Zakharov–Kuznetsov (mZK) model is a fundamental framework for describing nonlinear phenomena in plasma oscillations, optical systems, discrete electrical lattices, and oceanic wave dynamics. In this work, we establish a unified framework that combines exact solution construction with dynamical systems analysis of the mZK equation. Using the modified Khater method and the Sardar subequation technique under a traveling wave transformation, we derive new and more general families of solitary traveling wave solutions, including trigonometric, hyperbolic, rational, exponential–rational, and complex soliton forms. We show how free parameters govern the existence, shape, and stability of these solutions, with two- and three-dimensional MATLAB simulations illustrating their structures. Furthermore, we reformulate the mZK equation as a planar dynamical system and classify its qualitative features through phase portraits, identifying centers, saddles, and cusps. With the inclusion of periodic external forcing, the reduced system displays bifurcation and chaos, following a period-doubling route with multistability and sensitivity to parameter variations. To the best of our knowledge, this is the first study that simultaneously delivers exact soliton solutions and a complete nonlinear dynamics characterization of the mZK equation. This integrated approach provides a novel template for investigating broader classes of nonlinear evolution equations.</p>

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Chaotic transitions and multistable regimes in a nonlinear wave model through hybrid analytical–numerical techniques

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年2月21日

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  • Muhammad Iqbal,
  • Syeda Sarwat Kazmi,
  • Muhammad Bilal Riaz,
  • Muhammad Aziz Rehman

摘要

The modified Zakharov–Kuznetsov (mZK) model is a fundamental framework for describing nonlinear phenomena in plasma oscillations, optical systems, discrete electrical lattices, and oceanic wave dynamics. In this work, we establish a unified framework that combines exact solution construction with dynamical systems analysis of the mZK equation. Using the modified Khater method and the Sardar subequation technique under a traveling wave transformation, we derive new and more general families of solitary traveling wave solutions, including trigonometric, hyperbolic, rational, exponential–rational, and complex soliton forms. We show how free parameters govern the existence, shape, and stability of these solutions, with two- and three-dimensional MATLAB simulations illustrating their structures. Furthermore, we reformulate the mZK equation as a planar dynamical system and classify its qualitative features through phase portraits, identifying centers, saddles, and cusps. With the inclusion of periodic external forcing, the reduced system displays bifurcation and chaos, following a period-doubling route with multistability and sensitivity to parameter variations. To the best of our knowledge, this is the first study that simultaneously delivers exact soliton solutions and a complete nonlinear dynamics characterization of the mZK equation. This integrated approach provides a novel template for investigating broader classes of nonlinear evolution equations.