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

We study traveling pulses on a lattice and in a continuum where all pairs of particles interact, contributing to the potential energy. The interaction may be positive or negative, depending on the particular pair but overall is positive in a certain sense. For such an interaction kernel $J$ with unit integral (or sum), the operator 1/ε²[J∗u-u], with ∗ continuous or discrete convolution, shares some common features with the spatial second derivative operator, especially when ε is small. Therefore, the equation $u_{t t}$ - 1/ε²[J∗u-u] + f(u)=0 may be compared with the nonlinear Klein Gordon equation $u_{t t}$ - $u_{x x}$$ + f(u)=0$. If $f$ is such that the Klein-Gordon equation has supersonic traveling pulses, we show that the same is true for the nonlocal version, both the continuum and lattice cases.