<p>We study generalized Beltrami fields defined as solutions to the system<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1605_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">$ \operatorname{rot}^{n}A=\lambda A $</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1605_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$ \lambda $</EquationSource> </InlineEquation> is a function and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1605_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">$ A=(P,Q,R) $</EquationSource> </InlineEquation> is a vector-valued function of the variables <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1605_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$ (x,y,z) $</EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1605_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$ n\in{𝕅} $</EquationSource> </InlineEquation>.For <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1605_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$ \lambda=1 $</EquationSource> </InlineEquation> and arbitrary natural <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1605_Article_IEq7.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$ n $</EquationSource> </InlineEquation>,the system is reduced to a completely integrable form,with the result depending on the parity of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1605_Article_IEq7.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$ n $</EquationSource> </InlineEquation>.For <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1605_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$ n=1 $</EquationSource> </InlineEquation> and an arbitrary function <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1605_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$ \lambda $</EquationSource> </InlineEquation>,the system is also reduced to a completely integrable form.</p>

Generalized Beltrami Fields. Exact Solutions

期刊论文
发表日期：

2025年9月30日

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M. V. Neshchadim

摘要

We study generalized Beltrami fields defined as solutions to the system $ \operatorname{rot}^{n}A=\lambda A $ , where $ \lambda $ is a function and $ A=(P,Q,R) $ is a vector-valued function of the variables $ (x,y,z) $ , $ n\in{𝕅} $ .For $ \lambda=1 $ and arbitrary natural $ n $ ,the system is reduced to a completely integrable form,with the result depending on the parity of $ n $ .For $ n=1 $ and an arbitrary function $ \lambda $ ,the system is also reduced to a completely integrable form.