Instability of breathers in the topological discrete sine-Gordon system (Q2730685)

The author of this very interesting paper investigates the instability of breathers in the topological discrete sine-Gordon (TDSG) system which is defined on space-time \(\mathbb Z\times I\;R\). Space is a regular discrete lattice of spacing \(h\in (0,2]\) and spatial position is denoted by \(n\in \mathbb Z\). The time \(t\) is continuous. The field \(\psi_n(t)\) evolves according to the differential-difference equation NEWLINE\[NEWLINE\ddot \psi_n =(4-h^2) (2h)^{-2} \cos\psi_n (\sin\psi_{n+1}+\sin\psi_{n-1})(4+h^2) (2h)^{-2} \sin\psi_n (\cos\psi_{n+1}+\cos\psi_{n-1}).NEWLINE\]NEWLINE This tends to the usual sine-Gordon PDE \((x=nh)\) as \(h\to 0\) for \(\varphi =2\psi \). For \(h=2\) this equation reduces to \(\ddot \psi_n=(-1/2)\sin\psi_n(\cos\psi_{n+1}+\cos\psi_{n-1})\). Then the one-site breather is \(\Psi_n (t)=0\) for \(n\neq 0\) or \(\Psi_n (t)=\theta (t)\) for \(n=0\), respectively. Here \(\theta (t)\) is any \(T\)-periodic solution of the pendulum equation \(\ddot \theta +\sin\theta =0\). It is demonstrated that the breather solutions recently discovered in the weakly coupled TDSG system are spectrally unstable. This is in contrast with more conventional spatially discrete systems (oscillator chains), whose breathers are always spectrally stable at sufficiently weak coupling. In addition, it is discussed that while the TDSG system captures the dynamics of kinks and antikinks quite faithfully even for large \(h\), the same is not true of kink-antikink bound states, that is, breathers.