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<p>In this paper, the variable coefficients Broer-Kaup-Kupershmit (vcBKK) equations undergo analysis through the Lie symmetry method, leading to the determination of their infinitesimal generators. The optimal system of vcBKK equations are constructed with the help of Olver’s method. Based on the optimal system, a multiple set of (1+1)-dimensional reduced equations are obtained by performing a symmetric reduction on the (2+1)-dimensional vcBKK equations. Using <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10773_2025_6166_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\(\left( {G'/G} \right)\)</EquationSource> </InlineEquation> method, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10773_2025_6166_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\((1/G')\)</EquationSource> </InlineEquation> method and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10773_2025_6166_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">\((G'/{G^2})\)</EquationSource> </InlineEquation> method respectively to solve the reduced equations. Several novel exact solutions for the vcBKK equations are derived. The impact of the coefficient functions on the solutions are explored by choosing various forms for the coefficient functions. The dynamical behavior of the solutions are analyzed by studying the figures of the exact solutions with different parameters. Finally, the conservation laws for the vcBKK equations are derived using the new conservation theorem, further enriching the study of the vcBKK equations.</p>

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Lie Symmetry Analysis and Exact Solutions of the Variable Coefficients Broer-Kaup-Kupershmit Equations with Conservation Laws

  • 期刊论文

  • 发表日期：

    2025年10月14日

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  • Jinzhou Liu,
  • Xiangpeng Xin,
  • Zhaowen Yan

摘要

In this paper, the variable coefficients Broer-Kaup-Kupershmit (vcBKK) equations undergo analysis through the Lie symmetry method, leading to the determination of their infinitesimal generators. The optimal system of vcBKK equations are constructed with the help of Olver’s method. Based on the optimal system, a multiple set of (1+1)-dimensional reduced equations are obtained by performing a symmetric reduction on the (2+1)-dimensional vcBKK equations. Using \(\left( {G'/G} \right)\) method, \((1/G')\) method and \((G'/{G^2})\) method respectively to solve the reduced equations. Several novel exact solutions for the vcBKK equations are derived. The impact of the coefficient functions on the solutions are explored by choosing various forms for the coefficient functions. The dynamical behavior of the solutions are analyzed by studying the figures of the exact solutions with different parameters. Finally, the conservation laws for the vcBKK equations are derived using the new conservation theorem, further enriching the study of the vcBKK equations.