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<p>We construct a family of singly periodic entire solutions to the Allen–Cahn equation in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1915_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {R}}^{3}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>. Our approach relies on the relationship between this equation and the two-dimensional Toda lattice, an important integrable system. The level sets of these solutions closely approximate the graphs of suitable scaled lump solutions of the Toda lattice.</p>

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From Toda Lumps to Singly Periodic Solutions of Allen–Cahn Equation

  • 期刊论文

  • 发表日期：

    2025年2月6日

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  • Weizhao Liang,
  • Yong Liu,
  • Jianmin Yang

摘要

We construct a family of singly periodic entire solutions to the Allen–Cahn equation in \({\mathbb {R}}^{3}\) R 3 . Our approach relies on the relationship between this equation and the two-dimensional Toda lattice, an important integrable system. The level sets of these solutions closely approximate the graphs of suitable scaled lump solutions of the Toda lattice.