Orbital stability and instability of periodic wave solutions for the $$\phi ^4$$-model

Abstract

In this work we find explicit periodic wave solutions for the classical $$\phi ^4$$ -model, and study their corresponding orbital stability/instability in the energy space. In particular, for this model we find at least four different branches of spatially-periodic wave solutions, which can be written in terms of Jacobi elliptic functions. Two of these branches correspond to superluminal waves (speeds larger than the speed of light), the third-one corresponds to sub-luminal waves and the remaining one corresponds to stationary complex-valued waves. In this work we prove the orbital instability of real-valued sub-luminal traveling waves. Furthermore, we prove that under some additional hypothesis, complex-valued stationary waves as well as the real-valued zero-speed sub-luminal wave are all stable. This latter case is related (in some sense) to the classical Kink solution of the $$\phi ^4$$ -model.