Secondary bifurcation for a nonlocal Allen–Cahn equation

Kousuke Kuto; Tatsuki Mori; Tohru Tsujikawa; Shoji Yotsutani

doi:10.1016/j.jde.2017.04.010

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 journal › Article › peer-review

7 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 language	English
Pages (from-to)	2687-2714
Number of pages	28
Journal	Journal of Differential Equations
Volume	263
Issue number	5
DOIs	https://doi.org/10.1016/j.jde.2017.04.010
Publication status	Published - 2017 Sept 5
Externally published	Yes

Keywords

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

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jde.2017.04.010

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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",

note = "Publisher Copyright: {\textcopyright} 2017 Elsevier Inc.",

year = "2017",

month = sep,

day = "5",

doi = "10.1016/j.jde.2017.04.010",

language = "English",

volume = "263",

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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

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DO - 10.1016/j.jde.2017.04.010

M3 - Article

AN - SCOPUS:85018186231

SN - 0022-0396

VL - 263

SP - 2687

EP - 2714

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 5

ER -

Secondary bifurcation for a nonlocal Allen–Cahn equation

Abstract

Keywords

ASJC Scopus subject areas

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