E. V. Orlenko et al. / Natural Science 2 (2010) 1287-1291

Copyright © 2010 SciRes. OPEN ACCESS

1291

2

2

(1)()

2

Atgka

is the precession frequency with regard to the shift, and

2

() (0)Aa

qt qkttg

is the soliton coordinate.

Such integrable systems in which the free energy is

conserved are known as Manakov equations [17]. The

higher the velocity of a soliton, the lower the free energy

of system [17]; therefore, the soliton is formally unstable

(it is accelerated). However, the effect of other excita-

tions (solitary or ordinary spin waves), which is disre-

garded here, does not cause dissipation [17]. With an

additional inhomogeneous potential added to Eq.26, the

system becomes nonintegrable, allowing for the nontriv-

ial interaction of a soliton with the environment. Never-

theless, if the interaction potential varies slowly within

the scale of the soliton length k-1 , the variations (propa-

gations) in the soliton and potential correlate and the free

energy can be treated as an adiabatic invariant in this

case [15,16]. Thus, for the spin system considered above,

we will arrive at the pattern of nonlinear magnetic vor-

tices that transform into macroscopic vortices of a mag-

netic field in a antiferromagnetic chain system. It is im-

portant to note that we analyzed the case when only one

spin was flipped. Considering the flip of two, three, etc.

spins, we will generate a set of solitons with different

frequencies 1, 2, …,

n that differ in the ex-

change interaction constants.

4. CONCLUSIONS

The universal form of the exchange interaction Ham-

iltonian of the system of particles with an arbitrary spin

in the spin representation is developed from the first

principles. The Hamiltonian described the antiferro-

magnetic S1-chain contains the biquadratic term with

determined coefficient. The ground state energy of the

chain is more dependent from the exchange interaction

in the case of a long chain. The exchange interaction

makes such system antiferromagnetic and strong

long-order correlated. The presence of biquadratic term

in the Hamiltonian of the system gives rice to nonlinear

spin waves in the chain and, in particular, to a soliton.

The soliton is stable if the nonpoint potential varies

slowly within the soliton length. For the one-dimen-

sional S=1spin system, we will arrive at the pattern of

nonlinear magnetic vortices that bring the phase separa-

tion.

5. ACKNOWLEDGEMENTS

This work was supported by the Ministry of Education and Science

of the Russian Federation (state contract no. P2326, federal program

“Scientific and Pedagogical Staff of Innovative Russia for

2009–2013").

REFERENCES

[1] Ho, T.L. and Yip, L. (2000) Fragmented and single con-

densate ground ststes of spin-1 Bose gas. Physical Re-

view Letters, 84, 4031-4034.

[2] Haldane F.D.M. (1983) Phase diagrams of F=2 spinor

Bose-Einstein condensates. Physical Review A, 50, 1153.

[3] Haldane F.D.M. (1983) Physical Letters, 93A, 464.

[4] Ciobanu, C.V., Yip, S.K. and Ho, T.-L. (2000) Phase

diagrams of F=2 spinor Bose-Einstein condensates.

Physical Review A, 61, 1050-1056.

[5] Ho, T.-L. (1998) Bose-einstein condensate in optical

traps. Physical Review Letters, 81, 742-745.

[6] Orlenko, E. (2007) The universal Hamiltonian of the

exchange interaction for the system of particles with an

arbitrary spin. International Journal of Quantum Chem-

istry, 107, 2838-2843.

[7] Albuquerque, A., Hamer, F., Chris, J. and Jaan, O.(2009)

Quantum phase diagram and excitations for the

one-dimensional S=1 Heisenberg antiferromagnet with

single-ion anisotropy. Physical Review B, 79, 054412.

[8] Affleck, I., Kennedy, T., Lieb, E.H. and Tasaki, H. (1987)

Rigorous results on valence-bond ground states in anti-

ferromagnets. Physical Review Letters, 59, 799.

[9] Schollwöck, U., Jolicoeur, T. and Garel, T. (1996) Onset

of incommensurability at the valence-bond-solid point in

the S51 quantum spin chain. Physical Review B, 53,

3304.

[10] Orlenko, E., Mazets, I. and Matisov, B. (2003) Nonlin-

ear magnetic phenomena in the Bose-Einstein condensate.

Technical Physics, 48, 26-30.

[11] García-Ripoll, J.J., Martin-Delgado, M.A. and Cirac, J.I.

(2004) Implementation of spin hamiltonians in optical

lattices. Physical Review Letters, 93, 250405.

[12] Dalla Torre, E.G., Berg, E. and Altman, E. (2006). Hid-

den order in 1D bose insulators. Physical Review Letters,

97, 260401.

[13] den Nijs, M. and Rommelse, K. (1989) Preroughening

transitions in crystal surfaces and valence-bond phases in

quantum spin chains. Physical Review B, 40, 4709.

[14] Busch Th and Anglin J R, (2001) Dark-bright solitons in

inhomogeneous bose-einstein condensates. Physical Re-

view Letters, 87, 010401.

[15] Trillo, S., Wabnitz, S., Wright, E.M. and Stegeman, G.I.

(1988) Optical solitary waves induced by cross-phase

modulation. Optics Letters, 13, 871-873.

[16] Christodoulides, D.N. (1988) Black and white vector

solitons in weakly birefringeht optical fibers. Physical

Letter A, 132, 451-452.

[17] Manakov, S.V. (1974) On the integrality and stochastic in

the discrete dynamical systems. Zh. Éksp. Teor. Fiz., 67,

543-555.