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

N1 - Funding Information:
We thank Dr. Kouta Sekine (Toyo University, Japan) for his helpful advice. We also express our gratitude to anonymous referees for insightful comments. This work was supported by CREST, Japan, JST, Japan Grant Number JPMJCR14D4; and by JSPS KAKENHI, Japan Grant Number JP19K14601.
Funding Information:
We thank Dr. Kouta Sekine (Toyo University, Japan) for his helpful advice. We also express our gratitude to anonymous referees for insightful comments. This work was supported by CREST, Japan , JST, Japan Grant Number JPMJCR14D4 ; and by JSPS KAKENHI, Japan Grant Number JP19K14601 .
Publisher Copyright:
© 2021 The Authors

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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U2 - 10.1016/j.cam.2021.113708

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 -