Standard Nearest Neighbor Discretizations of Klein-Gordon Models Cannot Preserve Both Energy and Linear Momentum

Abstract

We consider nonlinear Klein-Gordon wave equations and illustrate that standard discretizations thereof (involving nearest neighbors) may preserve either standardly defined linear momentum or total energy but not both. This has a variety of intriguing implications for the ``non-potential'' discretizations that preserve only the linear momentum, such as the self-accelerating or self-decelerating motion of coherent structures such as discrete kinks in these nonlinear lattices.

5 pages, 2 figures

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• Discrete Klein-Gordon models with static kinks free of the Peierls-Nabarro potential
• Normal form for travelling kinks in discrete Klein-Gordon lattices
• Exact static solutions for discrete ${\varphi }^4$ models free of the Peierls-Nabarro barrier: Discretized first integral approach
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• Kinks in dipole chains
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