No Access

MOTION OF SPACE CURVES IN THREE-DIMENSIONAL MINKOWSKI SPACE , SO(2,1) SPIN EQUATION AND DEFOCUSING NONLINEAR SCHRÖDINGER EQUATION

GOPAL MUNIRAJA

Department of Mathematics, Bishop Cotton Women's Christian College, Bangalore-560027, India

Search for more papers by this author

and

M. LAKSHMANAN

Center for Nonlinear Dynamics, Bharathidasan University, Tiruchirapalli 620 024, India

Search for more papers by this author

https://doi.org/10.1142/S0219887810004701Cited by:12 (Source: Crossref)

Abstract

We consider the dynamics of moving curves in three-dimensional Minkowski space and deduce the evolution equations for the curvature and torsion of the curve. Next by mapping a continuous SO(2,1) Heisenberg spin chain on the space curve in , we show that the defocusing nonlinear Schrödinger equation(NLSE) can be identified with the spin chain, thereby giving a geometrical interpretation of it. The associated linear eigenvalue problem is also obtained in a geometrical way.

Keywords:

References

H. Hasimoto, J. Fluid. Mech. 51, 477 (1972), DOI: 10.1017/S0022112072002307. Web of Science, Google Scholar
M. Lakshmanan, Th. W. Ruijgork and C. J. Thompson, Physica A 84, 577 (1976), DOI: 10.1016/0378-4371(76)90106-0. Web of Science, Google Scholar
M. Lakshmanan, Phys. Lett. A 61, 53 (1977), DOI: 10.1016/0375-9601(77)90262-6. Web of Science, Google Scholar
L. A. Takhtajan, Phys. Lett. A 64, 235 (1977), DOI: 10.1016/0375-9601(77)90727-7. Web of Science, Google Scholar
V. E. Zakharov and L. A. Takhtajan, Theor. Math. Phys. 38, 17 (1979), DOI: 10.1007/BF01030253. Web of Science, Google Scholar
G. L. Lamb Jr., J. Math. Phys. 18, 1654 (1977), DOI: 10.1063/1.523453. Web of Science, Google Scholar
M. J. Ablowitz and P. A. Clarkson , Solitons, Nonlinear Evolution Equations and Inverse Scattering ( Cambridge University Press , Cambridge , 1992 ) . Google Scholar
M. Lakshmanan (ed.) , Solitons ( Springer-Verlag , Berlin , 1988 ) . Google Scholar
M. Lakshmanan, Phys. Lett. A 64, 353 (1978), DOI: 10.1016/0375-9601(78)90264-5. Web of Science, Google Scholar
M. Lakshmanan, J. Math. Phys. 20, 1667 (1979), DOI: 10.1063/1.524273. Web of Science, Google Scholar
J. Cieslinski, P. K. Gragert and A. Sym, Phys. Rev. Lett. 57, 1507 (1986), DOI: 10.1103/PhysRevLett.57.1507. Web of Science, Google Scholar
J. S. Langer and R. Perline, J. Nonlinear Sci. 1, 71 (1991), DOI: 10.1007/BF01209148. Google Scholar
S. Murugesh and M. Lakshmanan, Int. J. Bifur. Chaos 15, 51 (2005), DOI: 10.1142/S0218127405012004. Link, Web of Science, Google Scholar
V. E. Zhakarov and A. B. Shabat, Sov. Phys. JETP. 37, 823 (1973). Google Scholar
K. Nakayama, H. Segur and M. Wadati, Phys. Rev. Lett. 69, 2603 (1992), DOI: 10.1103/PhysRevLett.69.2603. Web of Science, Google Scholar
K. Nakayama, J. Phy. Soc. Japan. 67, 3031 (1998), DOI: 10.1143/JPSJ.67.3031. Web of Science, Google Scholar
Q. Ding, J. Phys. A 32, 5087 (1999), DOI: 10.1088/0305-4470/32/27/308. Google Scholar
N. Gürbüz, Int. J. Math. Math. Sci. 16, 2609 (2005). Google Scholar
W. Kühnel , Differential Geometry: Curves-Surfaces-Manifolds ( American Mathematical Society , Providence, RI , 2006 ) . Google Scholar
S. Vijayalakshmi, Nonlinear dynamics of higher dimensional spin systems, Ph.D Thesis, Centre for Nonlinear Dynamics, Bharathidasan university, Tiruchirappalli, Tamil Nadu, India (2004) . Google Scholar
M. Lakshmanan and S. Vijayalakshmi . Google Scholar
M. Lakshmananet al., J. Math. Phys. 39, 3765 (1998), DOI: 10.1063/1.532466. Web of Science, Google Scholar
R. Myrzakulovet al., J. Math. Phys. 39, 2122 (1998), DOI: 10.1063/1.532279. Web of Science, Google Scholar