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<p>In this paper, we investigate a (3+1)-dimensional nonlinear molecular-type evolution model using a neural network symbolic computation approach (NNSCA). The model extends classical long-wave equations to higher dimensions and offers a unified framework for describing nonlinear dispersive phenomena in optics, plasmas, and fluid systems. By reformulating the governing equation into bilinear form, NNSCA combined with symbolic computation is applied to construct exact analytical solutions. Several neural architectures, including single- and double-hidden-layer configurations, are calibrated to obtain lump, breather, and multi-soliton interaction solutions. The derived solution families demonstrate rich nonlinear behaviors, such as lump–soliton interactions, lump-two-stripe resonance, periodic lump–kink structures, and breather-type oscillations. Their spatiotemporal evolution is represented through three-dimensional surface plots, revealing key features including localization, energy exchange, and structural deformation characteristic of molecular-type excitations. These results demonstrate the effectiveness of NNSCA in capturing high-dimensional coherent structures and complex nonlinear superposition phenomena. The proposed framework not only enriches the solution space of the (3+1)-dimensional molecular-type model but also offers valuable insight into nonlinear wave dynamics with potential applications in nonlinear optics, hydrodynamics, and plasma physics.</p>

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Lump–stripe solitons and breather inelastic collision dynamics in a (3+1)-dimensional molecular-type evolution model via neural network symbolic computation

  • 期刊论文

  • 发表日期：

    2026年2月26日

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  • Mati ur Rahman,
  • Sonia Akram

摘要

In this paper, we investigate a (3+1)-dimensional nonlinear molecular-type evolution model using a neural network symbolic computation approach (NNSCA). The model extends classical long-wave equations to higher dimensions and offers a unified framework for describing nonlinear dispersive phenomena in optics, plasmas, and fluid systems. By reformulating the governing equation into bilinear form, NNSCA combined with symbolic computation is applied to construct exact analytical solutions. Several neural architectures, including single- and double-hidden-layer configurations, are calibrated to obtain lump, breather, and multi-soliton interaction solutions. The derived solution families demonstrate rich nonlinear behaviors, such as lump–soliton interactions, lump-two-stripe resonance, periodic lump–kink structures, and breather-type oscillations. Their spatiotemporal evolution is represented through three-dimensional surface plots, revealing key features including localization, energy exchange, and structural deformation characteristic of molecular-type excitations. These results demonstrate the effectiveness of NNSCA in capturing high-dimensional coherent structures and complex nonlinear superposition phenomena. The proposed framework not only enriches the solution space of the (3+1)-dimensional molecular-type model but also offers valuable insight into nonlinear wave dynamics with potential applications in nonlinear optics, hydrodynamics, and plasma physics.