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We study the existence, the stability and the non-degeneracy of normalizedstanding-waves solutions to a one dimensional non-linear Schr\"odingerequation. The non-linearity belongs to a class of algebraic functionsappropriately defined. We can show that for some of these non-linearities onecan observe the existence of degenerate minima, and the multiplicity ofpositive, radially symmetric minima having the same mass and the same energy.We also prove the stability of the ground-state and the stability of normalizedstanding-waves whose profile is a minimum of the energy constrained to themass.

Degeneracy and multiplicity of standing-waves of the one-dimensional non-linear Schr\"odinger equation for a class of algebraic non-linearities