Odd Periodic Waves for some Klein-Gordon Type Equations: Existence and Stability

In this paper, we establish the existence and stability properties of odd periodic waves related to the Klein-Gordon type equations, which include the well known ${\varphi }^4$ and ${\varphi }^6$ models. Existence of periodic waves is determined by using a general planar theory of ODE. The spectral analysis for the corresponding linearized operator is established using the monotonicity of the period map combined with an improvement of the standard Floquet theory. Orbital stability in the odd sector of the energy space is proved using exclusively the monotonicity of the period map. The orbital instability of explicit solutions for the ${\varphi }^4$ and ${\varphi }^6$ models is presented using the abstract approach in \cite{grillakis1}.