Cnoidal waves for the cubic nonlinear Klein–Gordon and Schrödinger equations

We establish orbital stability results for cnoidal periodic waves of the cubic nonlinear Klein–Gordon and Schrödinger equations in the energy space restricted to zero mean periodic functions. More precisely, on the one hand, we prove that the cnoidal waves of the cubic Klein–Gordon equation are orbitally unstable as a direct application of the theory developed by Grillakis, Shatah, and Strauss. On the other hand, we show that the cnoidal waves for the Schrödinger equation are orbitally stable by constructing a suitable Lyapunov functional restricted to the associated zero mean energy space. The spectral analysis of the corresponding linearized operators, restricted to the periodic Sobolev space consisting of zero mean periodic functions, is performed using the Floquet theory and a Morse Index Theorem.