Monotonicity of the period map for the equation \(-\varphi ''+\varphi -\varphi ^{k}=0\)

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