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<p>The Helmholtz equation (HE) is comprised of two partial differential equations that illustrated wave propagation in a medium, especially for situations where the wavelength is much smaller than the spatial extent of the system. In this paper, we consider a coupled nonlinear HE to study its complex dynamics and analytical solutions. We portray some qualitative features such as: Bifurcation analysis, complex chaos dynamics, return map, recurrence plots, synchronization analysis and Sparse identification of nonlinear dynamics via applying sparse regression techniques. For analytical solutions, we use newly introduced method, i.e, Kumar-Malik approach to study soliton solutions. All results are depicted via 3D and 2D plots to showcase their behaviors. The integration of analytical, numerical, and data-driven techniques highlights the versatility and depth of the coupled HE system, with implications for nonlinear optics, wave dynamics, and computational modeling.</p>

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Qualitative characteristics, sparse identification of nonlinear dynamics, and new analytical results of the coupled nonlinear Helmholtz equations

  • 期刊论文

  • 发表日期：

    2025年6月30日

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  • Peiluan Li,
  • Rui Qiao,
  • Changjin Xu,
  • Miaoran Yiao,
  • Yizhen Qu

摘要

The Helmholtz equation (HE) is comprised of two partial differential equations that illustrated wave propagation in a medium, especially for situations where the wavelength is much smaller than the spatial extent of the system. In this paper, we consider a coupled nonlinear HE to study its complex dynamics and analytical solutions. We portray some qualitative features such as: Bifurcation analysis, complex chaos dynamics, return map, recurrence plots, synchronization analysis and Sparse identification of nonlinear dynamics via applying sparse regression techniques. For analytical solutions, we use newly introduced method, i.e, Kumar-Malik approach to study soliton solutions. All results are depicted via 3D and 2D plots to showcase their behaviors. The integration of analytical, numerical, and data-driven techniques highlights the versatility and depth of the coupled HE system, with implications for nonlinear optics, wave dynamics, and computational modeling.