<p>Based on the method of adjoint orbits, the classification of soliton solutions for a centro-affine curvature flow is carried out. The nontrivial periodic solutions for this centro-affine curvature flow is the Jacobi elliptic sine function, and the closed soliton solutions are a family of transcendental curves <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2024_3089_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\displaystyle \textrm{x}_{p,\;q}$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mtext>x</mtext> <mrow> <mi>p</mi> <mo>,</mo> <mspace width="0.277778em" /> <mi>q</mi> </mrow> </msub> </mstyle> </math></EquationSource> </InlineEquation> having rotation index <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2024_3089_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\displaystyle q$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </math></EquationSource> </InlineEquation> and closing up in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2024_3089_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\displaystyle p$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </math></EquationSource> </InlineEquation> periods of its centro-affine curvature function <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2024_3089_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\displaystyle \kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2024_3089_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="203" /> </InlineMediaObject> <EquationSource Format="TEX">$\displaystyle q/p\in (0,\;1/2)\cup (1/2,\;+\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mi>q</mi> <mo stretchy="false">/</mo> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.277778em" /> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>,</mo> <mspace width="0.277778em" /> <mo>+</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2024_3089_Article_IEq6.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$\displaystyle q,\; p$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mi>q</mi> <mo>,</mo> <mspace width="0.277778em" /> <mi>p</mi> </mrow> </mstyle> </math></EquationSource> </InlineEquation> are relatively prime positive integers.</p>

The soliton solutions of a third-order centro-affine curvature flow

期刊论文
发表日期：

2025年1月16日

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Huidi Niu,
Yun Yang

摘要

Based on the method of adjoint orbits, the classification of soliton solutions for a centro-affine curvature flow is carried out. The nontrivial periodic solutions for this centro-affine curvature flow is the Jacobi elliptic sine function, and the closed soliton solutions are a family of transcendental curves $\displaystyle \textrm{x}_{p,\;q}$ $x_{p, q}$ having rotation index $\displaystyle q$ $q$ and closing up in $\displaystyle p$ $p$ periods of its centro-affine curvature function $\displaystyle \kappa $ $κ$ , where $\displaystyle q/p\in (0,\;1/2)\cup (1/2,\;+\infty )$ $q / p \in (0, 1 / 2) \cup (1 / 2, + \infty)$ and $\displaystyle q,\; p$ $q, p$ are relatively prime positive integers.