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In this paper we investigate the existence of solutions in $H^1(R^N) \timesH^1(R^N)$ for nonlinear Schr\"odinger systems of the form \[ \left\{\begin{aligned} -\Delta u_1 &= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1 +r_1\beta |u_1|^{r_1-2}u_1|u_2|^{r_2}, \\ -\Delta u_2 &= \lambda_2 u_2 + \mu_2|u_2|^{p_2 -2}u_2 + r_2 \beta |u_1|^{r_1}|u_2|^{r_2 -2}u_2, \end{aligned}\right. \] under the constraints \[\int_{R^N}|u_1|^2 \, dx = a_1>0,\quad\int_{R^N}|u_2|^2 \, dx = a_2>0. \] Here $ N \geq 1, \beta >0, \mu_i >0, r_i>1, 2 <p_i < 2 + \frac{4}{N}$ for $i=1,2$ and $ r_1 + r_2 < 2 + \frac{4}{N}$.This problem is motivated by the search of standing waves for an evolutionproblem appearing in several physical models. Our solutions are obtained asconstrained global minimizers of an associated functional. Note that in thesystem $\lambda_1$ and $\lambda_2$ are unknown and will correspond to theLagrange multipliers. Our main result is the precompactness of the minimizingsequences, up to translation, and as a consequence we obtain the orbitalstability of the standing waves associated to the set of minimizers.

Existence and orbital stability of standing waves for nonlinear Schr\"odinger systems