Dihedral molecular configurations interacting by Lennard-Jones and Coulomb forces

Irina Berezovik; Wieslaw Krawcewicz; Qingwen Hu

doi:10.3934/dcdss.2019124

Article Contents

2019, Volume 12, Issue 7: 1879-1903. Doi: 10.3934/dcdss.2019124

This issue Previous Article The Morse property for functions of Kirchhoff-Routh path type Next Article Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger Systems

Dihedral molecular configurations interacting by Lennard-Jones and Coulomb forces

1.
Department of Mathematical Sciences, University of Texas at Dallas, 800 W Campbell Road, Richardson, Texas 75080-3021, USA
2.
Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China
3.
University of Texas at Dallas, Department of Mathematical Sciences, 800 W Campbell Road, Richardson, Texas 75080-3021, USA
4.
Department of Mathematics and Computer Science, Alcorn State University, 1000 ASU Drive, Lorman, MS, 39096, USA

^* Corresponding author: Wieslaw Krawcewicz
^* Corresponding author: Wieslaw Krawcewicz

Received: November 30, 2017

Revised: June 30, 2018

Published: June 2019

The first and second authors are supported by the Department of Mathematical Sciences University of Texas at Dallas. The third author is supported by the Center for Applied Mathematics at Guangzhou University, Guangzhou China and the Department of Mathematical Sciences University of Texas at Dallas

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Abstract

In this paper, we investigate nonlinear periodic vibrations of a group of particles with a planar dihedral configuration governed by the Lennard-Jones and Coulomb forces. Using the gradient equivariant degree, we provide a full topological classification of the periodic solutions with both temporal and spatial symmetries. In the process, we provide general formulae for the spectrum of the linearized system of equations describing the above configuration, which allows us to obtain the critical frequencies of the particles' motions. The obtained frequencies represent the set of all critical periods for small amplitude periodic solutions emerging from a given stationary symmetric orbit of solutions.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Stationary solution to equation (7) with dihedral symmetries

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Figure 2. Relative motions of all 6 particles with $ \lambda_0^2 = \frac{l^2}{\mu} $, $ l = 1 $ and $ \mu $ near the eigenvalue $ \mu_0 = 10.10496819 $ of $ \nabla^2 V(u^o) $

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Figure 3. Relative motions of all particles with $ \lambda_0^2 = \frac{l^2}{\mu} $, $ l = 1 $ and $ \mu $ near the eigenvalue $ \mu = 6.442637681 $ of $ \nabla^2 V(u^o) $

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Figure 4. Relative motions of all particles with $ \lambda_0^2 = \frac{l^2}{\mu} $, $ l = 1 $ and $ \mu $ near the eigenvalue $ \mu = 8.469351217 $ of $ \nabla^2 V(u^o) $

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Figure 5. Relative motions of all particles with Relative motions of all particles with $\lambda_0^2=\frac{l^2}{\mu}$, $l=1$ and $\mu$ near the eigenvalue $\mu=3.854423919$ of $\nabla^2 V(u^o)$

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Table 1. The values $ \lambda_{j, l} $ in the critical set $ \Lambda $

$ j $	$ \mu_j $	$ \lambda_{j, 1} $	$ \lambda_{j, 2} $	$ \lambda_{j, 3} $	$ \lambda_{j, 4} $
0	10.10496819	0.31458103	0.62916205	0.94374308	1.25832410
1	8.469351217	0.34361723	0.68723445	1.03085168	1.37446891
3	3.854423919	0.50935463	1.01870927	1.52806390	2.03741854
$ 2^+ $	6.442637681	0.62767390	1.25534781	1.88302171	2.51069561
$ 2^- $	0.007288929	11.7130006	23.4260011	35.1390017	46.8520023

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Table 2. Maximal orbit types in $\mathscr W_{j,l}$

$\mathscr W_{j,l}$, $l\ge 1$	maximal orbit types
$\mathscr W_{0,l}$	$(D_6\times D_l)$
$\mathscr W_{1l}$	$({D_6}^{{\mathbb{Z}_1}}{ \times _{{D_6}}}{D_{6l}}) - ({D_2}^{{D_1}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}}) - ({D_2}^{{{\tilde D}_1}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}})$
$\mathscr W_{2,l}$	$({D_6}^{{\mathbb{Z}_2}}{ \times _{{D_3}}}{D_{3l}}) - ({D_2}^{{\mathbb{Z}_2}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}}) - ({D_2} \times {D_l})$
$\mathscr W_{3,l}$	$({D_6}^{{{\tilde D}_3}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}})$

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