Nonlinear instabilities of multi-site breathers in Klein-Gordon lattices (Q2822878)

The paper considers the discrete Klein-Gordon (KG) equationNEWLINENEWLINENEWLINE\[NEWLINE \ddot{u}_n + V'(u_n)=\varepsilon(u_{n+1}-2u_n+u_{n-1}), \,\;n \in \mathbb{Z} NEWLINE\]NEWLINENEWLINENEWLINE\noindent where \(V: \mathbb{R} \rightarrow \mathbb{R}\) is an on-site potential, \(\varepsilon>0\) is the coupling constant and \(u_n\) are the amplitudes of the coupled nonlinear oscillators with \(\{ u_n\}_{n \in \mathbb{Z}} \in \mathbb{R}^\mathbb{Z}\).NEWLINENEWLINEThere are studied the breather solutions of the aforementioned KG equation which are \(T\)-periodic in time and exponentially localized in space, i.e. they are given by the vector \(\mathbf{u}(t)=\mathbf{u}(t+T)\) such that \(\mathbf{u} \in \mathcal{C}_{\mathrm{per}}^\infty ((0, T; l^2(\mathbb{Z}))\) if \(V\) is smooth.NEWLINENEWLINEBy using asymptotic expansions, it is shown that ``the multi-site breathers are unstable in the KG lattice equation if the Krein signatures of the internal mode and of the wave spectrum are opposite to each other.''NEWLINENEWLINEThe analytical results are illustrated by numerical simulations for the case of two-site breathers and two types of on-site potentials: (a) the Morse potential (accounting for the class of soft potentials) \(V(u)=(\mathrm{e}^{-u} -1)^2 /2\) and (b) the \(\phi^4\) potential (accounting for the class of hard potentials) \(V(u)=u^2/2+u^4/4\).