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 -