As indicated in the introductory chapter, discrete breathers are one of the most prototypical excitations that can arise in nonlinear dynamical lattices. They are periodic in time and strongly localized in space (Flach, Gorbach in Phys Rep 467:1, 2008, Aubry in Physica D 216:1, 2006). In integrable systems such as the sine-Gordon equation, such solutions were long known (Dodd, Eilbeck, Gibbon, Morris in Solitons and nonlinear wave equations. Academic Press, San Diego, 1982, Cuevas, Kevrekidis, Williams in The sine-Gordon model and its applications: from pendula and Josephson junctions to gravity and high energy physics. Springer, Heidelberg, 2014) but were also believed to be rather special. However, a series of investigations by Sievers, Takeno, Page, and others (Sievers, Takeno in Rev Lett 61:970–973, 1988, Page in Phys Rev B 41:7835–7838, 1990) in the late 1980s and early 1990s suggested that they might be generic for nonlinear lattices of the Klein–Gordon (i.e., onsite nonlinearity) and FPUT (i.e., intersite nonlinearity) types. Then, the work of (MacKay, Aubry in Nonlinearity 7:1623, 1994) provided a mathematical foundation for this expectation illustrating that under (fairly generic) non-resonance conditions, discrete breathers should be expected to persist in such nonlinear, spatially extended discrete dynamical systems.
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