November  2019, 12(7): 1879-1903. doi: 10.3934/dcdss.2019124

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

Received  December 2017 Revised  July 2018 Published  December 2018

Fund Project: 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.

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.

Citation: Irina Berezovik, Wieslaw Krawcewicz, Qingwen Hu. Dihedral molecular configurations interacting by Lennard-Jones and Coulomb forces. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1879-1903. doi: 10.3934/dcdss.2019124
References:
[1]

Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Differential Equations & Dynamical Systems, Vol. 1, 2006.  Google Scholar

[2]

Z. Balanov, W. Krawcewicz and H. Ruan, Periodic solutions to O(2) × S1-symmetric variational problems: Equivariant gradient degree approach, Nonlinear Analysis and Optimization II. Optimization, 45-84, Contemp. Math., 514, Amer. Math. Soc., Providence, RI, 2010. doi: 10.1090/conm/514/10099.  Google Scholar

[3]

Z. BalanovW. KrawcewiczS. Rybicki and H. Steinlein, A short treatise on the equivariant degree theory and its applications, J. Fixed Point Theory App., 8 (2010), 1-74.  doi: 10.1007/s11784-010-0033-9.  Google Scholar

[4]

I. Berezovik, C. García-Azpeitia and W. Krawcewicz, Symmetries of nonlinear vibrations in tetrahedral molecular configurations, DCDS-B (accepted September 2018). Google Scholar

[5]

T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Springer-Verlag, New York-Berlin, 1985. doi: 10.1007/978-3-662-12918-0.  Google Scholar

[6]

M. Dabkowski, W. Krawcewicz, Y. Lv and H-P. Wu, Multiple Periodic Solutions for Γ-symmetric Newtonian Systems, J. Diff. Eqns., 10 (2017), 6684-6730. doi: 10.1016/j.jde.2017.07.027.  Google Scholar

[7]

T. tom Dieck, Transformation Groups., Walter de Gruyter, 1987. doi: 10.1515/9783110858372.312.  Google Scholar

[8]

J. Fura, A. Ratajczak and S. Rybicki, Existence and continuation of periodic solutions of autonomous Newtonian systems, J. Diff. Eqns, 218 (2005), 216-252. doi: 10.1016/j.jde.2005.04.004.  Google Scholar

[9]

C. Garcia-Azpeitia and J. Ize, Global bifurcation of polygonal relative equilibria for masses, vortices and dNLS oscillators, J. Diff. Eqns, 251 (2011), 3202-3227.  doi: 10.1016/j.jde.2011.06.021.  Google Scholar

[10]

C. Garcia-Azpeitia and M. Tejada-Wriedt, Molecular chains interacting by Lennard-Jones and Coulomb forces, Quali. Theory Dyn. Syst., 16 (2017), 591-608. doi: 10.1007/s12346-016-0221-0.  Google Scholar

[11]

K. Geba, Degree for gradient equivariant maps and equivariant Conley index, in Topological Nonlinear Analysis Ⅱ (Frascati, 1995), Progr. Nonlinear Differential Equations App., 27, Birkhäuser, Boston, 1997,247-272.  Google Scholar

[12]

A. Gołebiewska and S. Rybicki, Global bifurcations of critical orbits of G-invariant strongly indefinite functionals, Nonlinear Analysis, TMA, 74 (2011), 1823-1834.  doi: 10.1016/j.na.2010.10.055.  Google Scholar

[13]

E. Goursat, Sur les substitutions orthogonales et les divisions régulières de l'espace, Annales scientifiques de l'École Normale Supérieure, 6 (1889), 9-102. doi: 10.24033/asens.317.  Google Scholar

[14]

J. Ize and A. Vignoli, Equivariant Degree Theory, vol. 8 of De Gruyter Series in Nonlinear Analysis and Applications, Berlin, Boston: De Gruyter., 2003. doi: 10.1515/9783110200027.  Google Scholar

[15]

J. E. Lennard-Jones, On the determination of molecular fields, Proc. R. Soc. Lond. A, 106 (1924), 463-477. Google Scholar

[16]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Math. Sciences, Vol. 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[17]

K. H. Mayer, G-invariante Morse-Funktionen, Manuscripta Math., 63 (1989), 99-114. doi: 10.1007/BF01173705.  Google Scholar

[18]

H. Ruan and S. Rybicki, Applications of equivariant degree for gradient maps to symmetric Newtonian systems, Nonlinear Anal., 68 (2008), 1479-1516.  doi: 10.1016/j.na.2006.12.039.  Google Scholar

[19]

S. Rybicki, Applications of degree for S1-equivariant gradient maps to variational nonlinear problems with S1-symmetries, Topol. Methods Nonlinear Anal., 9 (1997), 383-417.  doi: 10.12775/TMNA.1997.018.  Google Scholar

[20]

E. H. Spanier, Algebraic Topology, McGraw-Hill Book Co, New York-Toronto-Lomdon, 1966.  Google Scholar

[21]

H.-P. Wu, GAP program for the computations of the Burnside ring A(Γ × O(2)), https://bitbucket.org/psistwu/gammao2, developed at University of Texas at Dallas, 2016. Google Scholar

[22]

, Symmetry Resources at Otterbein University, http://symmetry.otterbein.edu/gallery/. Google Scholar

show all references

References:
[1]

Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Differential Equations & Dynamical Systems, Vol. 1, 2006.  Google Scholar

[2]

Z. Balanov, W. Krawcewicz and H. Ruan, Periodic solutions to O(2) × S1-symmetric variational problems: Equivariant gradient degree approach, Nonlinear Analysis and Optimization II. Optimization, 45-84, Contemp. Math., 514, Amer. Math. Soc., Providence, RI, 2010. doi: 10.1090/conm/514/10099.  Google Scholar

[3]

Z. BalanovW. KrawcewiczS. Rybicki and H. Steinlein, A short treatise on the equivariant degree theory and its applications, J. Fixed Point Theory App., 8 (2010), 1-74.  doi: 10.1007/s11784-010-0033-9.  Google Scholar

[4]

I. Berezovik, C. García-Azpeitia and W. Krawcewicz, Symmetries of nonlinear vibrations in tetrahedral molecular configurations, DCDS-B (accepted September 2018). Google Scholar

[5]

T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Springer-Verlag, New York-Berlin, 1985. doi: 10.1007/978-3-662-12918-0.  Google Scholar

[6]

M. Dabkowski, W. Krawcewicz, Y. Lv and H-P. Wu, Multiple Periodic Solutions for Γ-symmetric Newtonian Systems, J. Diff. Eqns., 10 (2017), 6684-6730. doi: 10.1016/j.jde.2017.07.027.  Google Scholar

[7]

T. tom Dieck, Transformation Groups., Walter de Gruyter, 1987. doi: 10.1515/9783110858372.312.  Google Scholar

[8]

J. Fura, A. Ratajczak and S. Rybicki, Existence and continuation of periodic solutions of autonomous Newtonian systems, J. Diff. Eqns, 218 (2005), 216-252. doi: 10.1016/j.jde.2005.04.004.  Google Scholar

[9]

C. Garcia-Azpeitia and J. Ize, Global bifurcation of polygonal relative equilibria for masses, vortices and dNLS oscillators, J. Diff. Eqns, 251 (2011), 3202-3227.  doi: 10.1016/j.jde.2011.06.021.  Google Scholar

[10]

C. Garcia-Azpeitia and M. Tejada-Wriedt, Molecular chains interacting by Lennard-Jones and Coulomb forces, Quali. Theory Dyn. Syst., 16 (2017), 591-608. doi: 10.1007/s12346-016-0221-0.  Google Scholar

[11]

K. Geba, Degree for gradient equivariant maps and equivariant Conley index, in Topological Nonlinear Analysis Ⅱ (Frascati, 1995), Progr. Nonlinear Differential Equations App., 27, Birkhäuser, Boston, 1997,247-272.  Google Scholar

[12]

A. Gołebiewska and S. Rybicki, Global bifurcations of critical orbits of G-invariant strongly indefinite functionals, Nonlinear Analysis, TMA, 74 (2011), 1823-1834.  doi: 10.1016/j.na.2010.10.055.  Google Scholar

[13]

E. Goursat, Sur les substitutions orthogonales et les divisions régulières de l'espace, Annales scientifiques de l'École Normale Supérieure, 6 (1889), 9-102. doi: 10.24033/asens.317.  Google Scholar

[14]

J. Ize and A. Vignoli, Equivariant Degree Theory, vol. 8 of De Gruyter Series in Nonlinear Analysis and Applications, Berlin, Boston: De Gruyter., 2003. doi: 10.1515/9783110200027.  Google Scholar

[15]

J. E. Lennard-Jones, On the determination of molecular fields, Proc. R. Soc. Lond. A, 106 (1924), 463-477. Google Scholar

[16]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Math. Sciences, Vol. 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[17]

K. H. Mayer, G-invariante Morse-Funktionen, Manuscripta Math., 63 (1989), 99-114. doi: 10.1007/BF01173705.  Google Scholar

[18]

H. Ruan and S. Rybicki, Applications of equivariant degree for gradient maps to symmetric Newtonian systems, Nonlinear Anal., 68 (2008), 1479-1516.  doi: 10.1016/j.na.2006.12.039.  Google Scholar

[19]

S. Rybicki, Applications of degree for S1-equivariant gradient maps to variational nonlinear problems with S1-symmetries, Topol. Methods Nonlinear Anal., 9 (1997), 383-417.  doi: 10.12775/TMNA.1997.018.  Google Scholar

[20]

E. H. Spanier, Algebraic Topology, McGraw-Hill Book Co, New York-Toronto-Lomdon, 1966.  Google Scholar

[21]

H.-P. Wu, GAP program for the computations of the Burnside ring A(Γ × O(2)), https://bitbucket.org/psistwu/gammao2, developed at University of Texas at Dallas, 2016. Google Scholar

[22]

, Symmetry Resources at Otterbein University, http://symmetry.otterbein.edu/gallery/. Google Scholar

Figure 1.  Stationary solution to equation (7) with dihedral symmetries
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) $
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) $
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) $
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 $\nabla^2 V(u^o)$
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
$ 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
Table 2.  Maximal orbit types in $\mathscr W_{j,l}$
$\mathscr W_{j,l}$, $l\ge 1$ maximal orbit types
$\mathscr W_{0,l}$ $(D_6\times D_l)$
$\mathscr W_{1l}$ $({D_6}^{{\mathbb{Z}_1}}{ \times _{{D_6}}}{D_{6l}}) - ({D_2}^{{D_1}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}}) - ({D_2}^{{{\tilde D}_1}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}})$
$\mathscr W_{2,l}$ $({D_6}^{{\mathbb{Z}_2}}{ \times _{{D_3}}}{D_{3l}}) - ({D_2}^{{\mathbb{Z}_2}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}}) - ({D_2} \times {D_l})$
$\mathscr W_{3,l}$ $({D_6}^{{{\tilde D}_3}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}})$
$\mathscr W_{j,l}$, $l\ge 1$ maximal orbit types
$\mathscr W_{0,l}$ $(D_6\times D_l)$
$\mathscr W_{1l}$ $({D_6}^{{\mathbb{Z}_1}}{ \times _{{D_6}}}{D_{6l}}) - ({D_2}^{{D_1}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}}) - ({D_2}^{{{\tilde D}_1}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}})$
$\mathscr W_{2,l}$ $({D_6}^{{\mathbb{Z}_2}}{ \times _{{D_3}}}{D_{3l}}) - ({D_2}^{{\mathbb{Z}_2}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}}) - ({D_2} \times {D_l})$
$\mathscr W_{3,l}$ $({D_6}^{{{\tilde D}_3}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}})$
[1]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240

[2]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[3]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[4]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

[5]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[6]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[7]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273

[8]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[9]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[10]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[11]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[12]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[13]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (65)
  • HTML views (378)
  • Cited by (0)

Other articles
by authors

[Back to Top]