Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction (Q862091)

The authors consider the nonlocal wave equation \[ u_{tt}-\frac{1}{\varepsilon^2}(j_{\varepsilon}*u-u)+f(u) =0,\eqno(1) \] for \(t>0\) and \(x\in \mathbb{R}\), where \(*\) is a continuous or discrete convolution and \(f\) is a \(C^2\) function with \(f(0)=0\). The also consider a lattice version of \((1)\), i.e., \[ \ddot{u}_n- \frac{1}{\varepsilon^2}\sum_{k=-\infty}^{\infty}\alpha_ku_{n-k}+f(u_n)=0\eqno(2) \] for \(n\in \mathbb{Z}\), and they study the homoclinic traveling wave solutions of both \((1)\) and \((2)\), which decay at infinity.