of atoms with attractive behavior, as is the group of atoms investigated in [20]. With the above choice of parameters we obtain the solutions where A$_{10}$ = A$_{20}$ = A$_{0}$ = 2 n K (1 $_{/}$√$_{2)}$$_{/}$ √$_{π}$ , for n = 1 , 2 , ... and K (1 $_{/}$√$_{2)}$ being the elliptic integral. The value of A$_{0}$ is obtained through appropriate boundary conditions; see [20]. We use Eq. (12) to get ζ = √$_{π}$ 2 [1 + erf ( ξ )] . Thus, the pair of coupled solutions are given by with η being a real function obeying Eq. (13). Next, we use Eq. (16) to obtain the function χ ( t ) . As two distinct examples, we consider the cases f = 1 and f = 1 + ϵ cos( ω$_{0}$t ) , to describe periodic and quasiperiodic possibilities, respectively. The first case, f = 1 , leads us to χ = [1 + 15 cos $^{2}$(2 t )] 1 / $^{2}$/ 2 . The second case, f = 1+ ϵ cos( ω$_{0}$t ) , is a little more involved; it leads us to χ which is given as a combination of the two solutions of the Mathieu equation d $^{2}$z dt 2 + 4 f ( t ) z = 0 . It has the form χ = (2 z 2 1 + 2 z 2 $_{2}$/W $^{2}$) 1 / $^{2}$, where z$_{1}$$_{,}$$_{2}$ represent the two linearly independent solutions of the Mathieu equation, with W being the corresponding Wronskian. In Fig. 1 we show the the potential v ( x, t ) for the two cases, (a) periodic, and (b) quasiperiodic. The solution (20) is plotted in Fig. 2, for the periodic and quasiperiodic cases, respectively. The initial data for the Mathieu equation are z (0) = √$_{2}$ and dz/dt (0) = 0 . The results for ψ$_{2}$ are similar and we do not show them. To verify stability of the solutions, we use the split-step method, through finite difference to investigate the propagation of the initial states ψ$_{1}$ ( x, 0) and ψ$_{2}$ ( x, 0) given by Eqs. (20) and (21), at t = 0 , with the inclusion of a small aleatory perturbation of 3% on the profile of the initial states. With this procedure, we could verify that the solutions (20) and (21) are both stable against such aleatory perturbations. Example # 2 – In this case, we consider an attractive potential but we all ow the nonlinearity to change, as we did in Ref. [21]. We choose the nonlinearity terms to be, 5 g$_{jk}$ = G$_{jk}$χ − $^{1}$exp( ξ $^{2}$/γ$^{2}$) (22) ψ$_{1}$ ( x, t ) = e $_{iη}$A$_{0}$ exp( ξ $^{2}$/ 2) √$_{2}$ √ χ sn ( A$_{0}$ζ, 1 $_{/}$√$_{2)}$ dn ( A$_{0}$ζ, 1 , (20) ψ$_{2}$ ( x, t ) = e $_{iη}$A$_{0}$ exp( ξ $^{2}$/ 2) 2 √ χ sn ( A$_{0}$ζ, 1 $_{/}$√$_{2)}$ dn ( A$_{0}$ζ, 1 , (21) A$_{1}$ ( x, t ) = A$_{10}$ sn ( A$_{10}$ζ, 1 $_{/}$√$_{2)}$ √$_{2}$$_{dn}$$_{(}$$_{A}$$_{10}$$_{ζ,}$$_{1}$$_{/}$√$_{2)}$ , (18) A$_{2}$ ( x, t ) = A$_{20}$ sn ( A$_{20}$ζ, 1 $_{/}$√$_{2)}$ 2 dn ( A$_{20}$ζ, 1 , (19)