Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry

Sanjay Dharmavaram; Timothy J. Healey

doi:10.3934/dcdss.2019112

Article Contents

2019, Volume 12, Issue 6: 1669-1684. Doi: 10.3934/dcdss.2019112

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Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry

Sanjay Dharmavaram^1, , and
Timothy J. Healey^2,

1.
Department of Mathematics, 1 Dent Dr, Bucknell University, Lewisburg, PA 17837, USA
2.
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA

* Corresponding author: Sanjay Dharmavaram
* Corresponding author: Sanjay Dharmavaram

Received: April 30, 2018

Revised: May 31, 2018

Published: April 2019

The work of TJH was supported in part by the National Science Foundation through grant DMS-1613753, which is gratefully acknowledged

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Abstract

We consider bifurcation problems in the presence of $ O(3) $ symmetry. Well known group-theoretic techniques enable the classification of all maximal isotropy subgroups of $ O(3) $, with associated mode numbers $\ell∈\mathbb{N} $, leading to 1-dimensional fixed-point subspaces of the $ (2\ell+1) $-dimensional space of spherical harmonics. In each case the so-called equivariant branching lemma can then be used to establish the existence of a local branch of bifurcating solutions having the symmetry of the respective subgroup. To first-order, such a branch is a precise linear combination of the $ 2\ell+1 $ spherical harmonics, which we call the bifurcation direction. Our work here is focused on the direct construction of these bifurcation directions, complementing the above-mentioned classification. The approach is an application of a general method for constructing families of symmetric spherical harmonics, based on differentiating the fundamental solution of Laplace's equation in $ \mathbb{R}^3 $.

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Mathematics Subject Classification: Primary: 37G40, 58E09, 33C55; Secondary: 13A50.

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Figure 1. Regular tetrahedron

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Figure 2. A $\mathbb{T}$-invariant spherical harmonic for $\ell = 3$; (a) and (b) are diametrically opposite views

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Figure 3. $\mathbb{O}\oplus Z_2^c$-invariant spherical harmonic for (a) $\ell = 4$ and (b) $\ell = 6$

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Figure 4. (a) $\mathbb{O}$-invariant spherical harmonic for $\ell = 9$; (b) $\mathbb{O}^-$-invariant spherical harmonic for $\ell = 9$

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Figure 5. $\mathbb{I}\oplus Z_2^c$-invariant basis functions for (a) $\ell = 6$ and (b) $\ell = 10$

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Figure 6. $\mathbb{I}$-invariant spherical harmonic for $\ell = 15$

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Figure 7. $D_{6}^d$-invariant spherical harmonic of order $\ell = 3$: (a) Front view and (b) top view

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Figure 8. $D_4^d$-invariant spherical harmonic of order $\ell = 5$: (a) Front view and (b) top View

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Figure 9. One of the basis function that generate the two dimensional subspace of $\mathbb{D}_4\oplus Z_2^c$-invariant spherical harmonic of order $\ell = 4$: (a) Front view and (b) top view

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Figure 10. The two basis functions (a) and (b) that span the subspace of $\mathbb{O}$-invariant spherical harmonics or order $\ell = 12$

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Table 1. Subgroups of $O(3)$ and their invariant spherical harmonic basis. Here $s\in\{0, 1\}, p, q\in\mathbb{N}\cup\{0\}$

Group	Invariant Spherical Harmonic Basis	Order
$\mathbb{T}$	$\mathcal{T}^s_6 \mathcal{T}^p_4 \mathcal{T}^q_3(1/r)\vert_{r=1}$	$6s+4p+3q$
$\mathbb{O}$	$\mathcal{O}_9^s \mathcal{O}^p_6 \mathcal{O}^q_4(1/r)\vert_{r=1}$	$9s+6p+4q$
$\mathbb{I}$	$\mathcal{I}_{15}^s\mathcal{I}^p_{10}\mathcal{I}^q_{6}(1/r)\vert_{r=1}$	$15s+10p+6q$
$\mathbb{T}\oplus Z_2^c$	$\mathcal{T}_6^s \mathcal{T}^{2p}_3 \mathcal{T}^q_4(1/r)\vert_{r=1}$	$6s+6p+4q$
$\mathbb{O}\oplus Z_2^c$	$\mathcal{O}^p_6 \mathcal{O}^q_4(1/r)\vert_{r=1}$	$6p+4q$
$\mathbb{I}\oplus Z_2^c$	$\mathcal{I}^p_{10}\mathcal{I}^q_{6}(1/r)\vert_{r=1}$	$10p+6q$
$\mathbb{O}^{-}$	$\mathcal{T}_4^p\mathcal{T}_3^q(1/r)\vert_{r=1}$	$4p+3q$
$Z_n$	$\hat{z}^p \mathcal{C}_{qn}(1/r)\vert_{r=1}$, $\hat{z}^p \mathcal{S}_{qn}(1/r)\vert_{r=1}$	$p+qn$
$D_n$	$\hat{z}^{2p} \mathcal{C}_{qn}(1/r)\vert_{r=1}$, $\hat{z}^{2p+1} \mathcal{S}_{qn}(1/r)\vert_{r=1}$	$2p+qn$, $2p+1+qn$ (resp.)
$D_n^z$	$\hat{z}^{p}\mathcal{C}_{qn}(1/r)\vert_{r=1}$	$p+qn$
$Z_{2n}^-$ (even $n$), $Z_n\oplus Z_2^c$ (odd $n$)	$\hat{z}^{2p+j} \mathcal{C}_{qn}(1/r)\vert_{r=1}$, $\hat{z}^{2p+j} \mathcal{S}_{qn}(1/r)\vert_{r=1}$,	$2p+j+qn$
	where $j = qn (\text{ mod }2)$
$Z_{2n}^-$ (odd $n$), $Z_n\oplus Z_2^c$ (even $n$)	$\hat{z}^{2p} \mathcal{C}_{qn}(1/r)\vert_{r=1}$, $\hat{z}^{2p} \mathcal{S}_{qn}(1/r)\vert_{r=1}$	$2p+qn$
$D_{2n}^d$ (even $n$), $D_n\oplus Z_2^c$ (odd $n$)	$\hat{z}^{2p}\mathcal{C}_{2qn}(1/r)\vert_{r=1}$, $\hat{z}^{2p+1}\mathcal{S}_{(2q+1)n}(1/r)\vert_{r=1}$	$2p+2qn$, $2p+1+(2q+1)n$ (resp.)
$D_{2n}^d$ (odd $n$), $D_n\oplus Z_2^c$ (even $n$)	$\hat{z}^{2p}\mathcal{C}_{qn}(1/r)\vert_{r=1}$	$2p+qn$

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