<p>The Kairat equations play a significant role in modeling nonlinear phenomena in plasma physics and optical fibers. This study investigates the symmetry structure and integrability properties of the combined Kairat-II-X equation, which unifies the characteristics of both the Kairat-II and Kairat-X models. To analyze its integrability, we first perform a Painlevé analysis, followed by the Lie symmetry method to construct an optimal system of vector fields and obtain systematic similarity reductions. The resulting ordinary differential equations yield a variety of exact solutions, including multi-soliton and breather soliton forms. In addition, the Hirota bilinear method is employed to derive lump-type soliton solutions, further enriching the family of analytical solutions. Graphical representations are provided to illustrate the complex dynamical behavior of these solutions. The findings advance the theoretical understanding of the Kairat-II-X equation and extend existing studies by presenting new integrable structures and solution forms not previously reported in the literature.</p>

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Integrability and Dynamic Solutions of the Novel Unified Kairat-II-X Equation via Symmetry Methods

  • Sachin Kumar

摘要

The Kairat equations play a significant role in modeling nonlinear phenomena in plasma physics and optical fibers. This study investigates the symmetry structure and integrability properties of the combined Kairat-II-X equation, which unifies the characteristics of both the Kairat-II and Kairat-X models. To analyze its integrability, we first perform a Painlevé analysis, followed by the Lie symmetry method to construct an optimal system of vector fields and obtain systematic similarity reductions. The resulting ordinary differential equations yield a variety of exact solutions, including multi-soliton and breather soliton forms. In addition, the Hirota bilinear method is employed to derive lump-type soliton solutions, further enriching the family of analytical solutions. Graphical representations are provided to illustrate the complex dynamical behavior of these solutions. The findings advance the theoretical understanding of the Kairat-II-X equation and extend existing studies by presenting new integrable structures and solution forms not previously reported in the literature.