Abstract <p>In Dunkl theory on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10507_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\mathbb{R}}^{n}}\)</EquationSource>  </InlineEquation> which generalizes classical Fourier analysis, we study the solution of the Klein–Gordon equation defined by: <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10507_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="407" /> </InlineMediaObject> <EquationSource Format="TEX">\(\partial _{t}^{2}u - {{\Delta }_{k}}u = - {{m}^{2}}u,\;\;\;u(x,0) = g(x),\;\;\;{{\partial }_{t}}u(x,0) = f(x),\)</EquationSource>  </InlineEquation>with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10507_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(m > 0\)</EquationSource>  </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10507_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">\(\partial _{t}^{2}u\)</EquationSource>  </InlineEquation> is the second derivative of the solution <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10507_Article_IEq5.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(u\)</EquationSource>  </InlineEquation> with respect to <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10507_Article_IEq6.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">\(t\)</EquationSource>  </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10507_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\Delta }_{k}}u\)</EquationSource>  </InlineEquation> is the Dunkl Laplacian with respect to <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10507_Article_IEq8.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(x\)</EquationSource>  </InlineEquation> where <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10507_Article_IEq9.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(f\)</EquationSource>  </InlineEquation> and <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10507_Article_IEq10.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(g\)</EquationSource>  </InlineEquation> are the two functions in <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11982_2025_10507_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{S}({{\mathbb{R}}^{n}})\)</EquationSource>  </InlineEquation> which surround the initial conditions. We obtain an integral representation for its solution which we gives some properties. As a specific result, we studied the associated energies to the Dunkl–Klein–Gordon equation.</p>

On the Study of the Klein–Gordon Equation in the Dunkl Setting

期刊论文
发表日期：

2025年7月1日

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M. Gaidi,
M. Bedhiafi

摘要

Abstract

In Dunkl theory on \({{\mathbb{R}}^{n}}\) which generalizes classical Fourier analysis, we study the solution of the Klein–Gordon equation defined by: \(\partial _{t}^{2}u - {{\Delta }_{k}}u = - {{m}^{2}}u,\;\;\;u(x,0) = g(x),\;\;\;{{\partial }_{t}}u(x,0) = f(x),\) with \(m > 0\) and \(\partial _{t}^{2}u\) is the second derivative of the solution \(u\) with respect to \(t\) and \({{\Delta }_{k}}u\) is the Dunkl Laplacian with respect to \(x\) where \(f\) and \(g\) are the two functions in \(\mathcal{S}({{\mathbb{R}}^{n}})\) which surround the initial conditions. We obtain an integral representation for its solution which we gives some properties. As a specific result, we studied the associated energies to the Dunkl–Klein–Gordon equation.