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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

Peter Bates 1, and  Chunlei Zhang 2,

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.
Keywords: nonlocal wave equation, nonlinear pulses., Klein-Gordon lattice.
Mathematics Subject Classification: Primary: 34K30, 37L60; Secondary: 34C37, 82B2.
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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