  In this paper, we establish the monotonicity of the period map in terms of
the energy levels for certain periodic solutions of the equation
$-\varphi''+\varphi-\varphi^{k}=0$, where $k>1$ 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.
