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We prove that standing-waves solutions to the non-linear Schr\"odingerequation in dimension one whose profiles can be obtained as minima of theenergy over the mass, are orbitally stable and non-degenerate, provided thenon-linear term $ G $ satisfies a Euler differential inequality. When thenon-linear term $ G $ is a combined pure power-type, then there is only onepositive, symmetric minimum of prescribed mass.

Orbital stability and uniqueness of the ground state for NLS in dimension one