The equilibrium phase diagram of the dilute

The equilibrium phase diagram of the dilute bose gas exhibits a continuous phase transition between condensed
(Na,Rb,Li) and Non condensed (He) phases1. One of the most difference between in these two systems deceit in their
mutual interaction either very weak in the former or very strong in the later2. The condensed phase vanishes above
some critical temperature Tcand raise continuously with decreasing temperature below this critical point. However ,
the dynamical process of condensate formation has proved to be challenging phenomenon of both theoretically and
experimentally. This formation process called Bose-Einstein Condensate(BEC).
The dilute gaseous of BEC is identical in weakly interacting gases are also modeled by the NonLinear
Schrondinger(NLS) equation with an external potential known as the Gross-Pitaevskii(GP) equation3.The nonlinearity
in the GP equation introduced by the inter atomic interactions. Since, the coefficient ‘g'(coupling constant)
can be controlled by the s-wave scattering length. The scattering length can be tuned by an external magnetic ,
optical or dc-electric field.The possibility of controlling the inter atomic interactions in BECs has motivated in many
experimental and theoretical studies3.
In 1998, Stamper-Kurn have strongly confined N a234andRb875 atoms in optical dipole trap.They achieved
BEC with internal degree of freedom corresponding to the three hyperfine spin state F=1,mf = +1, ?1, 06.In general,
the spinor BEC was experimentally achieved in an optical trap by the MIT group in a spin-1 N a23 BEC which was
found to have the antiferromagnetic ground state7.The same group subsequently demonstrate the formation of spin
domains8 and quantum tunneling across the spin domains9.The main feature of spinor BEC is that two or more
hyperfine states of the atoms in the condensate have almost the same energy. As a result of this spin degree of
freedom becomes a suitable dynamical variable,which gives growth to new physics that are not present the usual
single-component BEC,where the spins are effectively frozen10.
The multicomponent in spinor BEC is investigated in detail.For examples- different types of solitons, namely
dark soliton11,gap soliton12,bright soliton13,bright-dark complexes14,bright dark magnetic soliton15 and rogue
wave16 have been reported. The modulational instability of a continuous wave state with constant density in
spinor BEC model as investigated in17 and some exact solution were reported in 18.The full – time description
o the modulational instability development in the integrable spinor BEC model was given both numerically19 and
analytically20.The non- autonomous of GP system of soliton solution are reported in 21.Wadati and co-workers
are reported the three component GP equation is integrable using 2×2 matrix22.The matrix of NLS equation can be
using the non vanishing boundary condition was developed in 23 and extended in 24. The bright soliton of spinor
BEC model shows two spin states such as ferromagnetic(nonzero total spin) and polar(zero total spin ) using inverse
scattering transformation method 22 .
The integrable models produce a very useful testing for new analytical and numerical approaches to study such a
difficult system as the spinor BEC. At the same time the integrability conditions introduce a special restrictions on
the parameter of the model which can conflict with actual experimental settings,although the fact that the effective
interaction between atoms in BEC can be tuned, to some extent,by the optically induced feshbach resonance25-26.
A specific integrable version of this model to generate the exact soliton solution from a continuous wave( CW) state
using a Darboux transformation method27. The recent achievement in spinor BEC is generating of rogue wave in
Spinor BEC using a Dorboux dressing method16. In our work, we will concentrate on the generalized vector rational
solution of three component GP equation and search the condition for existing of bright soliton solution based on the
gauge transformation method.