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Fractional Dynamics pp 153–214Cite as

Fractional Dynamics of Media with Long-Range Interaction

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Part of the book series: Nonlinear Physical Science ((NPS,volume 0))

Abstract

Dynamics with long-range interaction has been the subject of continuing interest in different areas of science. The long-range interactions have been studied in discrete systems as well as in their continuous analogues. Models of classical spins with long-range interactions were studied in Refs. (Dyson, 1969a, b, 1971; Joyce, 1969; Frohlich et al., 1978; Nakano et al., 1994a,b,c, 1995; Sousa, 2005). An infinite 1-dimensional Ising model with long-range interactions was considered by Dyson (Dyson, 1969a,18b, 1971). The d-dimensional classical Heisenberg model with long-range interaction was described in Refs. (Joyce, 1969; Frohlich et al., 1978), and their quantum generalization can be found in Refs. (Nakano et al., 1994a, b, c). Kinks in the Frenkel-Kontorova model with long-range interparticle interactions were studied in Ref. (Braun et al., 1990). Solitons in a 1-dimensionallattice with the long-range Lennard-Jones-type interaction were considered in Ref. (Ishimori, 1982). The properties of time periodic spatially localized solutions (breathers) on discrete chains in the presence of algebraically decaying interactions were considered in Refs. (Flach, 1998; Gorbach and Flach, 2005). Energy and decay properties of discrete breathers in systems with long-range interactions were also studied in the framework of the Klein-Gordon equation (Braun and Kivshar, 1998; Flach, 1998; Baesens and MacKay, 1999; Braun and Kivshar, 2004), and discrete nonlinear Schrodinger equations (Gaididei et al., 1997,995; Mingaleev et al., 1998,2000; Rasmussen et al., 1998). Synchronization of chaotic systems with long-range interactions was discussed in (Tessone et al., 2006).

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    Vasily E. Tarasov

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Tarasov, V.E. (2010). Fractional Dynamics of Media with Long-Range Interaction. In: Fractional Dynamics. Nonlinear Physical Science, vol 0. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14003-7_8

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