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These equations control the functions ρ ( x, t ) , η ( x, t ) and ζ ( x, t ) , and the two potentials are given by 2 2 

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We take ζ ( x, t ) = F ( ξ ( x, t )) , with ξ ( x, t ) = x/χ ( t ) , which has to obey the Eqs. (7), (8) and (9). After some algebraic calculations we get that 

We use Eq. (9) to obtain 

Also, using Eq. (8) we can write 

where a = a ( t ) is a function of time. 

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The above results can be used to find finite energy solutions fo r both ψ$_{1}$ and ψ$_{2}$ , with appropriate choices of F ( ξ ) , according to Eq. (12). We illustrate these results with some applications of current interest to BEC. 

Example # 1. – The first example is a extension of the model discussed in [20 ], in the case of nonlinearities given by 

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with χ = χ ( t ) > 0 . We consider the simpler case, with µ$_{1}$ = µ$_{2}$ = 0 . Here, we get that v$_{1}$ ( x, t ) = v$_{2}$ ( x, t ) = v ( x, t ) , with v ( x, t ) being an attractive potential. We use Eq. (11) to get ρ ( x, t ) = exp( ξ $^{2}$/ 2) / √ χ and the potential is now given by 

with 

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With the choice a = ∫ χ − $^{2}$dt , we can write h ( t ) = 0 . Thus, the potentials acquire the simple form v ( x, t ) = f ( t ) x $^{2}$. Here we choose G$_{11}$ = − 1 / 2 , G$_{12}$ = − 1 , G$_{21}$ = − 1 / 2 , G$_{22}$ = − 1 , which we use in Eqs. (3-4). In this case, each one of the two solutions, ψ$_{1}$ and ψ$_{2}$ , describes a similar group 

