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

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

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