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<p>The Boussinesq type equation is a versatile nonlinear model for describing wave propagation in dispersive media, including shallow-water dynamics, tsunami evolution, and internal ocean waves. This study applies the Hirota bilinear method (HBM) to the compact form of the modified Boussinesq-type equation to construct an extensive family of exact analytical solutions, encompassing single- and multi-solitons, first- and second-order breathers, lump waves, and hybrid soliton–breather interactions. These solutions exhibit rich nonlinear phenomena such as phase shifts, amplitude modulation, resonance-induced fusion and fission, and long-range algebraic decay. High-resolution 3D, contour, and discrete visualizations confirm the spatiotemporal localization, oscillatory behaviors, and interaction patterns predicted by the analysis. Physically, the results correspond to real-world geophysical processes, including solitary tsunami fronts, modulated wave packets in coastal waters, and slowly decaying disturbances relevant to early tsunami detection. The findings expand the known solution space of the Boussinesq-type system and provide a rigorous mathematical framework for modeling complex wave phenomena in geophysical and environmental fluid contexts.</p>

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Resonant soliton, breather, and lump dynamics in the Boussinesq type equation via Hirota bilinear method: exact solutions and geophysical wave applications

  • 期刊论文

  • 发表日期：

    2026年4月16日

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  • N. Hemnath,
  • Sandip Saha,
  • Awani Bhushan

摘要

The Boussinesq type equation is a versatile nonlinear model for describing wave propagation in dispersive media, including shallow-water dynamics, tsunami evolution, and internal ocean waves. This study applies the Hirota bilinear method (HBM) to the compact form of the modified Boussinesq-type equation to construct an extensive family of exact analytical solutions, encompassing single- and multi-solitons, first- and second-order breathers, lump waves, and hybrid soliton–breather interactions. These solutions exhibit rich nonlinear phenomena such as phase shifts, amplitude modulation, resonance-induced fusion and fission, and long-range algebraic decay. High-resolution 3D, contour, and discrete visualizations confirm the spatiotemporal localization, oscillatory behaviors, and interaction patterns predicted by the analysis. Physically, the results correspond to real-world geophysical processes, including solitary tsunami fronts, modulated wave packets in coastal waters, and slowly decaying disturbances relevant to early tsunami detection. The findings expand the known solution space of the Boussinesq-type system and provide a rigorous mathematical framework for modeling complex wave phenomena in geophysical and environmental fluid contexts.