<p>In this article, a novel conformable bilinear neural network strategy is adopted for the construction of exact analytic soliton solutions related to the (3+1)-dimensional conformable time-fractional Zakharov–Kuznetsov equation, which is a fundamental equation in plasma physics and the study of nonlinear wave dynamics. For the first time, bilinearization of the (3+1)-dimensional conformable time-fractional Zakharov–Kuznetsov equation is derived in the conformable fractional derivative framework, which is then integrated with neural network architectures in order to capture the complex wave interactions in a faithful way. Using single-hidden-layer architectures of configurations 4-3-1 and 4-4-1, breather wave and lump interaction solutions are obtained and highlight localized energy concentrations and oscillatory properties of the system. By extending the analysis to the double-hidden-layer architectures 4-2-2-1, 4-2-3-1 and 4-3-2-1, periodic waves, rogue waves and bright–dark solitons are obtained. The proposed methodology goes beyond the conventional methodology by leveraging the approximation power of neural networks to efficiently solve the bilinearized system, resulting in closed-form expressions without numerical approximations. Moreover, the 3D graphs with density projections, 2D graphs and contour graphs of some solutions with different fractional effects are also presented. This study not only contributes to the theoretical framework of fractional soliton dynamics but also provides a basis for hybrid analytical–computational methods directed to solving high-dimensional nonlinear partial differential equations with potential applications for sophisticated wave propagation modeling.</p>

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Dynamics of soliton solutions for the (3+1)-dimensional CTFZK equation in plasma physics using an advanced neural networking approach

  • Muhammad Qasim,
  • Ahmad Shafee,
  • Fengping Yao,
  • Muhammad Zafarullah Baber

摘要

In this article, a novel conformable bilinear neural network strategy is adopted for the construction of exact analytic soliton solutions related to the (3+1)-dimensional conformable time-fractional Zakharov–Kuznetsov equation, which is a fundamental equation in plasma physics and the study of nonlinear wave dynamics. For the first time, bilinearization of the (3+1)-dimensional conformable time-fractional Zakharov–Kuznetsov equation is derived in the conformable fractional derivative framework, which is then integrated with neural network architectures in order to capture the complex wave interactions in a faithful way. Using single-hidden-layer architectures of configurations 4-3-1 and 4-4-1, breather wave and lump interaction solutions are obtained and highlight localized energy concentrations and oscillatory properties of the system. By extending the analysis to the double-hidden-layer architectures 4-2-2-1, 4-2-3-1 and 4-3-2-1, periodic waves, rogue waves and bright–dark solitons are obtained. The proposed methodology goes beyond the conventional methodology by leveraging the approximation power of neural networks to efficiently solve the bilinearized system, resulting in closed-form expressions without numerical approximations. Moreover, the 3D graphs with density projections, 2D graphs and contour graphs of some solutions with different fractional effects are also presented. This study not only contributes to the theoretical framework of fractional soliton dynamics but also provides a basis for hybrid analytical–computational methods directed to solving high-dimensional nonlinear partial differential equations with potential applications for sophisticated wave propagation modeling.