Orbital Stability of ground state solutions of coupled nonlinear Schr"odinger equations

Orbital stability property for weakly coupled nonlinear Schrodinger equations is investigated. Different families of orbitally stable standing waves solutions will be found, generated by different classes of solutions of the associated elliptic problem. In particular, orbitally stable standing waves can be generated by least action solutions, but also by solutions with one trivial component whether or not they are ground states. Moreover, standing waves with components propagating with the same frequencies are orbitally stable if generated by vector solutions of a suitable Schro ̈dinger weakly coupled system, even if they are not ground states.