Coupled nonlinear Schr¨odinger equations (CNLSE) are very important because they can be used to model a great variety of physical systems. Solitary waves in these equations are often called vector solitons in the literature since they are generically described by two-component wave functions [1, 2]. One of the simplest vector solitons is known as shape-preserving, self-localized solution of coupled nonlinear evolution equations [1]. The CNLSE may also model beam prop- agation inside crystals as well as water wave interactions, and in fiber communication system, such equations have been shown to govern pulse propagation along orthogonal polarization axes in nonlinear optical fibers and in wavelength-division-mul tiplexed systems [3]. 

In the case of Bose-Einstein condensates, [4, 5], a many-body description of the effect of Fesh- bach resonances in the mean field limit, requires the use of tw o coupled Gross-Pitaevskii equations [6, 7]. This results in a convenient way to account for the physics of the so-called super-chemistry, [8]. In the case of one-dimensional (1D) Bose-Einstein condensates (BECs), the presence of cigar-shaped traps introduces tight confinement in two transverse direct ions, while leaving the condensate almost free along the longitudinal axis. The realization of BEC of trapped atoms was experimentally achieved in [9–11]. Next, dark solitons in B EC were formed with repulsive $^{87}$Rb atoms in [12, 13], and bright solitons were generated for attractive $^{7}$Li atoms in [14, 15]. In the case of two-component condensates, the experimental generation has been achieved for different hyperfine states in rubidium atoms in a magnetic trap [16] andin sodium atoms in an optical trap [17]. 

When treationg a set of CNLSE’s, one usually deals with two or more coupled NLSE’s with constant coefficients. More recently, however, in Refs. [18 –20] the authors have shown how to deal with NLSE with varying cubic coefficient, and in [21] onehas extended the procedure to the case of cubic and quintic nonlinearities. With this motivation on mind, our goal in the present work is to deal with soliton solutions of the CNLSE where the potentials and nonlinearities are modulated in space and time. The procedure consists in choosing an Ansatz which transforms two CNLSE’s into a pair of CNLSE’s with constant coefficients , certainly much easier to solve. With this procedure we can then obtain analytic solutions for some specific choices of parameters, leading us to interesting localized coupled soliton solutions of the bright and/or dark type. We illustrate the procedure with several distinct possibilities: firstly, we consider the system with attractive potentials and localized nonlinearities. Next, we consider nonlinearities delocalized and then we make the potential to change periodically in time from attractive to repulsive behavior. 

The model we start with is described by two CNLSE’s, with interactions up to third-order. 

