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For the cubic Schr\"odinger system with trapping potentials in$\mathbb{R}^N$, $N\leq3$, or in bounded domains, we investigate the existenceand the orbital stability of standing waves having components with prescribed$L^2$-mass. We provide a variational characterization of such solutions, whichgives information on the stability through of a condition ofGrillakis-Shatah-Strauss type. As an application, we show existence ofconditionally orbitally stable solitary waves when: a) the masses are small,for almost every scattering lengths, and b) in the defocusing, weaklyinteracting case, for any masses.

Stable solitary waves with prescribed $L^2$-mass for the cubic Schr\"odinger system with trapping potentials