Stability of Breathers for a Periodic Klein-Gordon Equation

The existence of breather-type solutions, i.e., solutions that are periodic in time and exponentially localized in space, is a very unusual feature for continuum, nonlinear wave-type equations. Following an earlier work establishing a theorem for the existence of such structures, we bring to bear a combination of analysis-inspired numerical tools that permit the construction of such waveforms to a desired numerical accuracy. In addition, this enables us to explore their numerical stability. Our computations show that for the spatially heterogeneous form of the ϕ4 model considered herein, the breather solutions are generically unstable. Their instability seems to generically favor the motion of the relevant structures. We expect that these results may inspire further studies towards the identification of stable continuous breathers in spatially heterogeneous, continuum nonlinear wave equation models.