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<p>The exact wave structures of a novel integrable <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((3+1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-dimensional wave equation related to shallow water wave phenomena are investigated. The Bell polynomial approach is employed to construct the Hirota bilinear form, Bäcklund transformation, and Lax pair of the model. Exact solutions, including two-wave, mixed lump–kink, transformed breather molecule, and solitary wave structures, are derived by using different analytical techniques. In addition, the Lie symmetry method is applied to reduce the governing equation to an ordinary differential equation, for which solitary wave solutions are obtained through the generalized logistic equation framework. The graphical features of the obtained solutions are illustrated through three-dimensional, contour, and line plots.</p>

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Analytical Characterization of Nonlinear Shallow Water Wave Dynamics in (3+1) Dimensions: Solitary and Periodic Structures

  • 期刊论文

  • 发表日期：

    2026年6月23日

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  • Fozia Bashir Farooq,
  • Nauman Raza,
  • Salman Tahir,
  • Melike Kaplan,
  • Muhammad Shahbaz

摘要

The exact wave structures of a novel integrable \((3+1)\) ( 3 + 1 ) -dimensional wave equation related to shallow water wave phenomena are investigated. The Bell polynomial approach is employed to construct the Hirota bilinear form, Bäcklund transformation, and Lax pair of the model. Exact solutions, including two-wave, mixed lump–kink, transformed breather molecule, and solitary wave structures, are derived by using different analytical techniques. In addition, the Lie symmetry method is applied to reduce the governing equation to an ordinary differential equation, for which solitary wave solutions are obtained through the generalized logistic equation framework. The graphical features of the obtained solutions are illustrated through three-dimensional, contour, and line plots.