Abstract
A XY Heisenberg spin chain model with two perpendicular spins par site is mapped onto a Kirchhoff thin elastic rod. It is shown that in the case of constant curvature the Euler—Lagrange equation leads to the static sine-Gordon equation. The case of a double-helical DNA-like configuration corresponds to two interacting Heisenberg spin chains and the corresponding Euler—Lagrange equation gives a system of coupled static sine-Gordon-type equations. The kink-antikink type and periodical static solutions for these models are derived. The soliton dynamics and the the nonlinear excitations of the systems are investigated. The interplay between curvature and nonlinear excitations is analyzed as well.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L. V. Yakushevich, Nonlinear Physics of DNA, Wiley-VCH, Weinheim (2004)
R. Dandoloff R. Balakrishnan, Quantum Effective Potential, Electron Transport an Conformons in Byopolimers, J. Phys. A: Math. Gen. 38, 6121–6127 (2005).
M. Daniel and V. Vasumathi, Perturbes Soliton Excitations in Inhomogeneous DNA, Phys. Rev. E (In press).
A. F. Fonseca and M. A. M. de Aguiar, Near equilibrium dynamics of nonhomogeneous Kirchhoff filaments in viscous media, Phys. Rev. E 63 016611 (2000)
I. Klapper, Biolgical Applications of the Dynamics of Twisted Elastic Rods, J. Comp. Phys. 125 325–337 (1996)
W. R. Bauer, R. A. Lund, and J. H. White, Twist and writhe of a DNA loop containing intrinsic bends, Proc. Natl. Acad. Sci. USA 90 833–837 (1993)
D. Bensimon, A. J. Simon, V. Croquette, and A. Bensimon, Stretching DNA with a receding meniscus: Experiments and models, Phys. Rev. Lett. 74 4754–4757 (1995)
Ph. Cluzel, A. Lebrun, Ch. Heller, R. Lavery, J.-L. Viovy, D. Chatenay, and F. Caron, DNA: An Extensible Molecule, Science 271 792–794 (1996) S. B. Smith, Y. Cui, and C. Bustamante, Overstretching B-DNA: The Elastic Response of Individual Double-Stranded and Single-Stranded DNA Molecules, ibid. 271 795–799 (1996) T. R. Strick, J.-F. Allemand, D. Bensimon, and V. Croquette, The Elasticity of a Single Supercoiled DNA Molecule, ibid. 271 1835–1837 (1996)
A. E H. Love, A Treatise on the Mathematical Theory od Elasticity, Dover Publications, New York (1944)
J. E. Marsden and Th. J. R. Hughes, Mathematical Foundations of Elasticity, Dover Publications, New York (1994)
A. Goriely and M. Tabor, Nonlinear dynamics of filaments I: Dynamical instabilities, Physica D 105 20–44 (1997); II: Nonlinear analysis, ibid. 105 45–61 (1997); III: Instabilities of helical rods, Proc. Roy. Soc. 453 2583–2601 (1997); IV: Spontaneous looping of elastic rods, ibid. 455 3183–3202 (1998)
M. Nizette and A. Goriely, Towards a classification of Euler-Kirchhoff filaments, J. Math. Phys. 40 2830–2866 (1999)
A. Goriely and M. Nizette, Kovalevskaya rods and Kovalevskaya waves, Regul. Chaotic Dyn. 45 no. 1, 95–106 (2000)
B. D. Coleman, E. H. Dill, M. Lembo, Zh. Lu and I. Tobias, On the Dynamics of Rods in the Theory of Kirchhoff and Clebsch, Arch. Rational Mech. Anal. 121 339–359 (1993)
S. S. Antman and T.-P. Liu, Travelling waves in hyperelastic rods, Quart. Appl. Math. 36 no. 4, 377–399 (1978/79)
M. Gaudin, La fonction d’onde de Bethe, Collection du Commissariat a ‘l’E’nergie Atomique: Se’rie Scientifique, Masson, Paris (1983)
K. M. Leung, Path integral approach to the statistical mechanics of solitons, Phys. Rev. B 26, 226–244 (1983); Mechanical properties of double-sine-Gordon solitons and the application to anisotropic Heisenberg ferromagnetic chains, Phys. Rev. B 27, 2877–2888 (1983).
M. Takahashi, Thermodynamics of One-Dimensional Solvable Models, Cambrigde University Press, Cambridge (2005)
H. B. Thacker, Exact Integrability in Quantum Field Theorry and Statistical Systems, Rev. Mod. Phys. 53 253–285 (1981)
R. Dandoloff and A. Saxena, Interaction induced deformation of two coupled XY spin chains, J.Phys.:Condens.Matter 9, L667–L670 (1997)
R. Dandoloff and A. Saxena, Nonlinear Sigma Model and the origin of geometric frustration on curved manifolds, Z. Phys. B 104 661–668 (1997)
M. Lakshmanan and S. Rajasekar, Nonlinear Dynamics: Integrability, Chaos and Patterns (Advanced texts in Physics), Springer-Verlag, Berlin (2003)
L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Course of Theoretical Physics, Vol 7), Pergamon Press, Oxford (1986)
T. McMillen and A. Goriely, Tendril Perversion in Intrinsically Curved Rods, J. Nonlin. Sci. 12, 241–281 (2002)
R. Balakrishnan and R. Dandoloff, The Schrödinger equation as a moving curve, Phys.Lett. A 260, 62–67 (1999)
N. G. Hunt and J. E. Hearst, Elastic model of DNA supercoiling in the infinite length limit, J. Chem. Phys. 12, 9329–9336 (1991)
F. Tanaka and H. Takahashi, Elastic theory of supercoiled DNA, J. Chem. Phys. 83, 6017–6026 (1985)
Y. Yang, I. Tobias, and W. K. Olson, Finite element analysis of DNA supercoiling, J. Chem. Phys. 98, 1673–1686 (1993)
T. P. Westcott, I. Tobias, and W. K. Olson, Modeling self-contact forces in the elastic theory of DNA supercoiling, J. Chem. Phys. 107, 3967–3980 (1997)
Y. Shi, A. E. Borivik and J. E. Hearst, Elastic rod model incorporating shear and extension, generalized Nonlinear Schrödinger equations, and novel closed-form solutions for supercoiled DNA, J. Chem. Phys. 103, 3166–3183 (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer
About this paper
Cite this paper
Grahovski, G.G., Dandoloff, R. (2007). A XY Spin Chain Models on Space Curves and Analogy with Kirchhoff Rods. In: Taş, K., Tenreiro Machado, J.A., Baleanu, D. (eds) Mathematical Methods in Engineering. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5678-9_34
Download citation
DOI: https://doi.org/10.1007/978-1-4020-5678-9_34
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-5677-2
Online ISBN: 978-1-4020-5678-9
eBook Packages: EngineeringEngineering (R0)