content_image/369102.jpg content_image/369103.jpg content_image/369101.jpg content_image/369100.jpg content_image/369099.jpg content_image/369098.jpg content_image/369097.jpg content_image/369096.jpg content_image/369095.jpg
These equations are given by 

<img src='content_image/369102.jpg'>

<img src='content_image/369103.jpg'>

where ψ$_{k}$ = ψ$_{k}$ ( x, t ) , the functions, v$_{k}$ ( x, t ) , are the trapping potentials, and the g$_{ij}$, ( i, j = 1 , 2) describe the strength of the cubic nonlinearities. Note that we are using standard notation, with both the fields and coordinates dimensionless. Note also tha t we are supposing that our model describe 1D BEC, and then it should engender cigar-shaped traps to induce tight confinement in two transverse directions, leaving the condensate almost free along the x axis. In the super- chemistry model of [7, 8], the above equations are amended by a term which goes as √$_{2}$$_{αψ}$$_{1}$$_{ψ}$$_{2}$ in Eq. (1) and with a term of α √$_{2}$ ψ 2 1 in Eq. (2). These, number-nonconserving, terms arise from treating the Feshbach resonance as a coupling term between atoms and molecules. 

It is interesting to comment here that the elimination of, say, ψ$_{2}$ in favor of an effective equation for ψ$_{1}$ , would necessarily end up with a GP equation with depletion effect. This could appear in the form of complex nonlinearity. We leave such study for a future work. 

In the following we seek to reduce the above two equations into the following pair of equations 

<img src='content_image/369101.jpg'>

<img src='content_image/369100.jpg'>

with µ$_{i}$ and the G$_{ij}$ being constants. We achieve this with the use of the two Anzatse : 

However, we now have to have 

<img src='content_image/369099.jpg'>

<img src='content_image/369098.jpg'>

<img src='content_image/369097.jpg'>

<img src='content_image/369096.jpg'>

<img src='content_image/369095.jpg'>

