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<p>The Benjamin–Bona–Mahony (BBM) equation is used to model unidirectional wave propagation in dispersive media, which describes wave behavior in physical systems where dispersion plays a significant role. This paper considers the extended version of the BBM equation in (3+1)-dimension. It has been noted that the considered equation was not studied when the fractal calculus is utilized. Here, in this work, the fractal form of the considered equation is taken into the consideration using the He’s fractal operator on semi-domain. The variational direct method, which integrates variational theory and the Ritz-like technique, is used in conjunction with the fractal based two scale transformation to develop exact solutions in semi-domain for considered equation. The obtained solutions are displayed via 3D simulations. The use of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1255_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\( \varrho = \frac{\ln 2}{\ln 3} \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϱ</mi> <mo>=</mo> <mfrac> <mrow> <mo>ln</mo> <mn>2</mn> </mrow> <mrow> <mo>ln</mo> <mn>3</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> and other values introduces fractal element into the solution, contributing to complex dynamics observed in plot. This complexity is the reason of fractal effects on the solution and showcases how different parameters’ values can can result in varied wave behaviors of the proposed model. The results obtained for the proposed model has potential applications in fluid dynamics and oceanography. It can model waves in the coastal environments having fractal-like characteristics.</p>

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Novel Fractal Soliton Solutions of a (3+1)-Dimensional Benjamin–Bona–Mahony Equation on a Cantor Set

  • 期刊论文

  • 发表日期：

    2025年3月22日

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  • M. M. Alqarni,
  • Emad E. Mahmoud,
  • M. A. Aljohani,
  • Shabir Ahmad

摘要

The Benjamin–Bona–Mahony (BBM) equation is used to model unidirectional wave propagation in dispersive media, which describes wave behavior in physical systems where dispersion plays a significant role. This paper considers the extended version of the BBM equation in (3+1)-dimension. It has been noted that the considered equation was not studied when the fractal calculus is utilized. Here, in this work, the fractal form of the considered equation is taken into the consideration using the He’s fractal operator on semi-domain. The variational direct method, which integrates variational theory and the Ritz-like technique, is used in conjunction with the fractal based two scale transformation to develop exact solutions in semi-domain for considered equation. The obtained solutions are displayed via 3D simulations. The use of \( \varrho = \frac{\ln 2}{\ln 3} \) ϱ = ln 2 ln 3 and other values introduces fractal element into the solution, contributing to complex dynamics observed in plot. This complexity is the reason of fractal effects on the solution and showcases how different parameters’ values can can result in varied wave behaviors of the proposed model. The results obtained for the proposed model has potential applications in fluid dynamics and oceanography. It can model waves in the coastal environments having fractal-like characteristics.