Research ArticlesNo Access

Anholonomy according to three formulations of non-null curve evolution

Nevin Gürbüz

Department of Mathematics and Computer Science, Eskişehir Osmangazi University, Eskişehir, Turkey

https://doi.org/10.1142/S0219887817501754Cited by:25 (Source: Crossref)

Abstract

In this paper, we present three anholonomy densities and three geometric phases for three formulations of curve evolution in Minkowski 3-space and give applications of three formulation for ferromagnetic and antiferromagnetic spin chains.

Keywords:

AMSC: 53Z05, 81Q70

References

1. M. V. Berrry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A Math. Phys. Sci. 392 (1984) 45–57. Crossref, Web of Science, Google Scholar
2. Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev. 115 (1959) 485–490. Crossref, Web of Science, Google Scholar
3. Y. Aharonov and J. Anandan, Phase change during a cylic quantum evolution, Phys. Rev. Lett. 58 (1987) 1593–1596. Crossref, Web of Science, Google Scholar
4. A. Tomita and R. Y. Chiao, Observation of Berry’s topological phase by use of an optical fiber, Phys. Rev. Lett. 57 (1986) 937–940. Crossref, Web of Science, Google Scholar
5. G. L. Lamb, Solitons on moving space curves, J. Math. Phys. 18 (1977) 1654. Crossref, Web of Science, Google Scholar
6. H. Hasimoto, Soliton on a vortex filament, J. Fluid Mech. 51 (1972) 477–485. Crossref, Web of Science, Google Scholar
7. H. Hasimoto, Motion of a vortex filament and its relation to elastica, J. Phys. Soc. Jpn. 31 (1971) 293–294. Crossref, Google Scholar
8. M. Lakshmanan, Continuum spin system as an exactly solvable dynamical system, Phys. Lett. A 61 (1977) 53–54. Crossref, Web of Science, Google Scholar
9. M. Lakshmanan, R. Myrzakulov, S. Vijayalakshmi and A. K. Danlybaeva, Motion of curves and surfaces and nonlinear evolution equations in $(2 + 1)$ $(2 + 1)$ dimensions, J. Math. Phys. 39 (1998) 3765–3771. Crossref, Web of Science, Google Scholar
10. Q. Ding and J. Inoguchi, Schrodinger flows, binormal motion for curves and the second AKNS-hierarchies, Chaos Solitons Fractals 21 (2004) 669–677. Crossref, Web of Science, Google Scholar
11. N. Gürbüz, The differential formula of Hasimoto transformation in Minkowski 3-space, Int. J. Math. Math. Sci. 16 (2005) 2609–2616. Crossref, Google Scholar
12. M. Kugler and S. Shtrikman. Berry’s phase, locally inertial frames, and classical analogues. Phys. Rev. 37 (1988) 934–937. Google Scholar
13. R. Dandoloff and W. Zakrzewski, Parallel transport along a space curve and related phases, J. Phys. A Math. Gen. 22 (1989) 461–466. Crossref, Google Scholar
14. R. Balakrishnan, R. Bishop and R. Dandoloff, Geometric phase in the classical continuous antiferromagnetic Heisenberg spin chain, Phys. Rev. Lett. 64 (1990) 2107–2110. Crossref, Web of Science, Google Scholar
15. R. Balakrishnan, A. R. Bishop and R. Dandoloff, Anholonomy of a moving space curve all applications to classical magnetic chains, Phys. Rev. B 47 (1993) 3108–3117. Crossref, Web of Science, Google Scholar
16. R. Balakrishnan and R. Dandoloff, The Schrödinger equation as a moving curve, Phys. Lett. A 260 (1999) 62–67. Crossref, Web of Science, Google Scholar
17. S. Murugesh and R. Balakrishnan, New connections between moving curves and soliton equations, Phys. Lett. A 290 (2001) 81–87. Crossref, Web of Science, Google Scholar
18. N. Gürbüz, Moving non-null curves according to Bishop frame in Minkowski 3-space, Int. J. Geom. Methods Mod. Phys. 12 (2015) 1550052. Link, Web of Science, Google Scholar
19. Z. W. Zhong, L. M. Li, Q. Y. Hai and W. Ke, Modified Heisenberg ferromagnet model and integrable equation, Commun. Theory. Phys. 44 (2005) 415–418. Crossref, Web of Science, Google Scholar
20. A. T. Ali, Moving non-Light like curves in Minkowski 3-space and soliton equations, Scitech J. Sci. Technol. 2 (2013) 17–25. Google Scholar
21. G. Munijara and M. Lakshmanan, Motion of space curves in three dimensional Minkowski space $ℝ_{1}^{3},$ SO(2,1) spin equation and defocusing nonlinear Schrödinger equation, Int. J. Geom. Methods Mod. Phys. 07 (2010) 1043. Link, Web of Science, Google Scholar
22. J. Langer and R. Perline, Local geometric invariants of integrable evolution equations, J. Math. Phys. 35 (1994) 1732–1737. Crossref, Web of Science, Google Scholar
23. S. Murugesh and M. Lakshmanan, Nonlinear dynamics of moving curves and surfaces: Application to physical systems, Int. J. Bifur. Chaos Appl. Sci. Eng. 15 (2005) 51–63. Link, Web of Science, Google Scholar
24. R. Balakrishnan and R. Blumenfeld, Transformation of general curve evolution to a modified Belavin-Polyakov equation, J. Math. Phys. 38 (1997) 5878–5888. Crossref, Web of Science, Google Scholar
25. R. Y. Chiao, Optical manifestation of Berry’s topological phases: Aharonov-Bohm-like effects for the photon, in Proc. 3rd Int. Symp. Foundations of Quantum Mechanic, Tokyo (1989), pp. 80–92. Google Scholar
26. A. Mostafazadeh, Relativistic adiabatic approximation and geometric phase, J. Phys. A Math. Gen. 31 (1998) 7829–7845. Crossref, Google Scholar
27. B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity (Academic Press, 1983). Google Scholar