Numerical verification for asymmetric solutions of the Hénon equation on bounded domains

Taisei Asai; Kazuaki Tanaka; Shin'ichi Oishi

doi:10.1016/j.cam.2021.113708

Numerical verification for asymmetric solutions of the Hénon equation on bounded domains

Taisei Asai^*, Kazuaki Tanaka, Shin'ichi Oishi

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

The Hénon equation, a generalized form of the Emden equation, admits symmetry-breaking bifurcation for a certain ratio of the transverse velocity to the radial velocity. Therefore, it has asymmetric solutions on a symmetric domain even though the Emden equation has no asymmetric unidirectional solution on such a domain. We discuss a numerical verification method for proving the existence of solutions of the Hénon equation on a bounded domain. By applying the method to a line-segment domain and a square domain, we numerically prove the existence of solutions of the Hénon equation for several parameters representing the ratio of transverse to radial velocity. As a result, we find a set of undiscovered solutions with three peaks on the square domain.

Original language	English
Article number	113708
Journal	Journal of Computational and Applied Mathematics
Volume	399
DOIs	https://doi.org/10.1016/j.cam.2021.113708
Publication status	Published - 2022 Jan 1

Keywords

Elliptic boundary value problem
Hénon equation
Numerical verification
Symmetry-breaking bifurcation

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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10.1016/j.cam.2021.113708

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title = "Numerical verification for asymmetric solutions of the H{\'e}non equation on bounded domains",

abstract = "The H{\'e}non equation, a generalized form of the Emden equation, admits symmetry-breaking bifurcation for a certain ratio of the transverse velocity to the radial velocity. Therefore, it has asymmetric solutions on a symmetric domain even though the Emden equation has no asymmetric unidirectional solution on such a domain. We discuss a numerical verification method for proving the existence of solutions of the H{\'e}non equation on a bounded domain. By applying the method to a line-segment domain and a square domain, we numerically prove the existence of solutions of the H{\'e}non equation for several parameters representing the ratio of transverse to radial velocity. As a result, we find a set of undiscovered solutions with three peaks on the square domain.",

keywords = "Elliptic boundary value problem, H{\'e}non equation, Numerical verification, Symmetry-breaking bifurcation",

author = "Taisei Asai and Kazuaki Tanaka and Shin'ichi Oishi",

note = "Publisher Copyright: {\textcopyright} 2021 The Authors",

year = "2022",

month = jan,

day = "1",

doi = "10.1016/j.cam.2021.113708",

language = "English",

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TY - JOUR

T1 - Numerical verification for asymmetric solutions of the Hénon equation on bounded domains

AU - Asai, Taisei

AU - Tanaka, Kazuaki

AU - Oishi, Shin'ichi

PY - 2022/1/1

Y1 - 2022/1/1

N2 - The Hénon equation, a generalized form of the Emden equation, admits symmetry-breaking bifurcation for a certain ratio of the transverse velocity to the radial velocity. Therefore, it has asymmetric solutions on a symmetric domain even though the Emden equation has no asymmetric unidirectional solution on such a domain. We discuss a numerical verification method for proving the existence of solutions of the Hénon equation on a bounded domain. By applying the method to a line-segment domain and a square domain, we numerically prove the existence of solutions of the Hénon equation for several parameters representing the ratio of transverse to radial velocity. As a result, we find a set of undiscovered solutions with three peaks on the square domain.

AB - The Hénon equation, a generalized form of the Emden equation, admits symmetry-breaking bifurcation for a certain ratio of the transverse velocity to the radial velocity. Therefore, it has asymmetric solutions on a symmetric domain even though the Emden equation has no asymmetric unidirectional solution on such a domain. We discuss a numerical verification method for proving the existence of solutions of the Hénon equation on a bounded domain. By applying the method to a line-segment domain and a square domain, we numerically prove the existence of solutions of the Hénon equation for several parameters representing the ratio of transverse to radial velocity. As a result, we find a set of undiscovered solutions with three peaks on the square domain.

KW - Elliptic boundary value problem

KW - Hénon equation

KW - Numerical verification

KW - Symmetry-breaking bifurcation

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DO - 10.1016/j.cam.2021.113708

M3 - Article

AN - SCOPUS:85110354857

SN - 0377-0427

VL - 399

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

M1 - 113708

ER -

Numerical verification for asymmetric solutions of the Hénon equation on bounded domains

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Keywords

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