Secondary bifurcation for a nonlocal Allen–Cahn equation

Kousuke Kuto, Tatsuki Mori, Tohru Tsujikawa, Shoji Yotsutani

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

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
Volume263
Issue number5
DOIs
Publication statusPublished - 2017 Sep 5
Externally publishedYes

Fingerprint

Allen-Cahn Equation
Nonlocal Equations
Bifurcation Point
Bifurcation
Bifurcation Curve
Elliptic integral
Global Bifurcation
Computer simulation
Neumann Problem
Level Set
Symmetry Breaking
Uniqueness
Odd
Symmetry
Numerical Simulation
Interval

Keywords

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

ASJC Scopus subject areas

  • Analysis

Cite this

Secondary bifurcation for a nonlocal Allen–Cahn equation. / Kuto, Kousuke; Mori, Tatsuki; Tsujikawa, Tohru; Yotsutani, Shoji.

In: Journal of Differential Equations, Vol. 263, No. 5, 05.09.2017, p. 2687-2714.

Research output: Contribution to journalArticle

Kuto, Kousuke ; Mori, Tatsuki ; Tsujikawa, Tohru ; Yotsutani, Shoji. / Secondary bifurcation for a nonlocal Allen–Cahn equation. In: Journal of Differential Equations. 2017 ; Vol. 263, No. 5. pp. 2687-2714.
@article{8e2b3d3827c849c2b2d15cb38487886b,
title = "Secondary bifurcation for a nonlocal Allen–Cahn equation",
abstract = "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.",
keywords = "Allen–Cahn equation, Bifurcation, Complete elliptic integrals, Nonlocal term, Symmetry breaking",
author = "Kousuke Kuto and Tatsuki Mori and Tohru Tsujikawa and Shoji Yotsutani",
year = "2017",
month = "9",
day = "5",
doi = "10.1016/j.jde.2017.04.010",
language = "English",
volume = "263",
pages = "2687--2714",
journal = "Journal of Differential Equations",
issn = "0022-0396",
publisher = "Academic Press Inc.",
number = "5",

}

TY - JOUR

T1 - Secondary bifurcation for a nonlocal Allen–Cahn equation

AU - Kuto, Kousuke

AU - Mori, Tatsuki

AU - Tsujikawa, Tohru

AU - Yotsutani, Shoji

PY - 2017/9/5

Y1 - 2017/9/5

N2 - 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.

AB - 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.

KW - Allen–Cahn equation

KW - Bifurcation

KW - Complete elliptic integrals

KW - Nonlocal term

KW - Symmetry breaking

UR - http://www.scopus.com/inward/record.url?scp=85018186231&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85018186231&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2017.04.010

DO - 10.1016/j.jde.2017.04.010

M3 - Article

AN - SCOPUS:85018186231

VL - 263

SP - 2687

EP - 2714

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 5

ER -