ORBITAL STABILITY AND UNIQUENESS OF THE GROUND STATE FOR THE NON-LINEAR SCHRODINGER EQUATION IN DIMENSION ONE

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