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In this paper, we prove existence and orbital stability results of periodicstanding waves for the cubic-quintic nonlinear Schr\"odinger equation. We usethe implicit function theorem to construct a smooth curve of explicit periodicwaves with \textit{dnoidal} profile and such construction can be used to provethat the associated period map is strictly increasing in terms of the energylevels. The monotonicity is also useful to obtain the behaviour of thenon-positive spectrum for the associated linearized operator around the wave.Concerning the stability, we prove that the dnoidal waves are orbitally stablein the energy space restricted to the even functions.

Periodic waves for the cubic-quintic non-linear Schr\"odinger equation: existence and orbital stability