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March  2006, 16(1): 235-252. doi: 10.3934/dcds.2006.16.235

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

 1 Department of Mathematics, Michigan State University, D211 Wells Hall , East Lansing, MI 48824, United States 2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, United States

Received  August 2005 Revised  January 2006 Published  June 2006

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.
Citation: Peter Bates, Chunlei Zhang. Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 235-252. doi: 10.3934/dcds.2006.16.235
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