Monotonicity of the period map for the equation -φ′′+φ-φk=0

In this paper, we establish the monotonicity of the period map in terms of the energy levels for certain periodic solutions of the equation -φ′′+φ-φk=0$$-\varphi ''+\varphi -\varphi ^{k}=0$$, where k>1$$k>1$$ is a real number. We present a new approach to demonstrate this property, utilizing spectral information of the corresponding linearized operator around the periodic solution and tools related to Floquet theory.