Lattice ODEs have a continuous time but a discrete
space variable, and can hence be regarded as discrete PDEs.
Despite their seeming simplicity they possess a very complex dynamics
and behave sometimes completely different than their continuous
counterparts. In this talk we discuss simple 1D Hamiltonian lattices and
present an overview on typical questions and problems that arise
when passing to a continuum scale. In particular, we emphasize the
role of strong oscillations and outline the connections to
modulation theory, nonlinear conservation laws, and thermodynamics.
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