Remarks on the orbital stability for the sine-Gordon equation

In this paper, we consider the problem of well-posedness and orbital stability of odd periodic traveling waves for the sine-Gordon equation. We first establish novel results concerning the local well-posedness in smoother periodic Sobolev spaces to guarantee the existence of a local time where the associated Cauchy problem has a unique solution with the zero mean property. Afterwards, we prove the orbital stability of odd periodic waves using a convenient index theorem applied to the constrained linearized operator defined in the Sobolev space with the zero mean property.