Secondary bifurcation for a nonlocal Allen–Cahn equation

Kousuke Kuto*, Tatsuki Mori, Tohru Tsujikawa, Shoji Yotsutani

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


This paper studies the Neumann problem of a nonlocal Allen–Cahn equation in an interval. A main result finds a symmetry breaking (secondary) bifurcation point on the bifurcation curve of solutions with odd-symmetry. Our proof is based on a level set analysis for the associated integral map. A method using the complete elliptic integrals proves the uniqueness of secondary bifurcation point. We also show some numerical simulations concerning the global bifurcation structure.

Original languageEnglish
Pages (from-to)2687-2714
Number of pages28
JournalJournal of Differential Equations
Issue number5
Publication statusPublished - 2017 Sept 5
Externally publishedYes


  • Allen–Cahn equation
  • Bifurcation
  • Complete elliptic integrals
  • Nonlocal term
  • Symmetry breaking

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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