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<p>This paper presents a training-free symbolic bilinear framework that couples Hirota’s bilinear method with bilinear-neural-network–inspired ansatz construction to derive exact wave solutions of the generalized (2+1)-dimensional Hirota–Satsuma–Ito (GHSI) equation. By extending the conventional linear neuron inputs to composite and partially nonlinear activation designs within both single- and dual-hidden-layer topologies, the proposed workflow systematically enlarges the trial-function space while keeping all parameters solvable through symbolic computation. As a result, we obtain several new families of hybrid interaction waves, including lump–two-kink interactions and periodic–kink interaction waves (with purely periodic and purely kink patterns arising as limiting cases), as well as richer lump–stripe interaction scenarios. The framework further reconstructs the standard Hirota N-soliton solution in a unified form and provides explicit examples for N=2, 3, 4, illustrating typical elastic interaction features via spatiotemporal visualizations. To interpret the irregular multi-scale modulation observed in representative patterns, we also perform a Duffing-type diagnostic on extracted solution time series and report chaos-like phase-portrait behavior at the solution level. These results enrich the solution library of the GHSI model and offer a systematic route for exploring complex nonlinear wave events relevant to shallow-water and coastal-wave dynamics.</p>

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Diverse Interaction Structures and Complex Wave Dynamics in the Generalized Hirota–Satsuma–Ito Equation

  • 期刊论文

  • 发表日期：

    2026年4月20日

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  • Jinxin Wu,
  • Junjie Li,
  • Chuyu Huang,
  • Yan Zhu,
  • Xiaohui Sun,
  • Runfa Zhang

摘要

This paper presents a training-free symbolic bilinear framework that couples Hirota’s bilinear method with bilinear-neural-network–inspired ansatz construction to derive exact wave solutions of the generalized (2+1)-dimensional Hirota–Satsuma–Ito (GHSI) equation. By extending the conventional linear neuron inputs to composite and partially nonlinear activation designs within both single- and dual-hidden-layer topologies, the proposed workflow systematically enlarges the trial-function space while keeping all parameters solvable through symbolic computation. As a result, we obtain several new families of hybrid interaction waves, including lump–two-kink interactions and periodic–kink interaction waves (with purely periodic and purely kink patterns arising as limiting cases), as well as richer lump–stripe interaction scenarios. The framework further reconstructs the standard Hirota N-soliton solution in a unified form and provides explicit examples for N=2, 3, 4, illustrating typical elastic interaction features via spatiotemporal visualizations. To interpret the irregular multi-scale modulation observed in representative patterns, we also perform a Duffing-type diagnostic on extracted solution time series and report chaos-like phase-portrait behavior at the solution level. These results enrich the solution library of the GHSI model and offer a systematic route for exploring complex nonlinear wave events relevant to shallow-water and coastal-wave dynamics.