Discrete Breathers with Dissipation
The interplay between discreteness and nonlinearity leads to the emergence of a new class of nonlinear excitations, viz. discrete breathers. These time-periodic and spatially localized excitations correspond to generic exact solutions of the underlying no
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Abstract The interplay between discreteness and nonlinearity leads to the emergence of a new class of nonlinear excitations, viz. discrete breathers. These time-periodic and spatially localized excitations correspond to generic exact solutions of the underlying nonlinear lattice models. Discrete breathers are not confined to certain lattice dimensions, nor are they sensitive to the particular type of nonlinearity in the system. They are usually dynamically and structurally stable and emerge in a variety of physical systems, ranging from lattice vibrations and magnetic excitations in crystals to light propagation in photonic structures and cold atom dynamics in periodic optical traps. Basic properties of discrete breathers, including spatial localization and stability, are briefly discussed in this chapter. Special focus is placed on a subclass of dissipative discrete breathers. Dissipation eliminates extended waves and allows for various resonances of discrete breathers with damped cavity modes. We discuss applications of the discrete breather concept in systems where dissipation is not only unavoidable but essential in order to observe and manipulate discrete breathers, and in order to use them for spectroscopic tools, amongst others.
1 Introduction This chapter is about localized excitations in spatially extended discrete systems, i.e. lattices. These systems are translationally invariant, implying the absence of disorder and defects. The common expectation – throw a stone into the water of a S. Flach Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Str. 38, D-01187 Dresden, Germany, [email protected] A.V. Gorbach Centre for Photonics and Photonic Materials, Department of Physics, University of Bath, Bath BA2 7AY, UK Flach, S., Gorbach, A.V.: Discrete Breathers with Dissipation. Lect. Notes Phys. 751, 289–320 (2008) c Springer-Verlag Berlin Heidelberg 2008 DOI 10.1007/978-3-540-78217-9 11
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lake and follow the evolution of the localized surface wave perturbation – is that an initially localized excitation would distribute its energy over the entire system in the course of time. What could stop such a delocalization process? It needs just two ingredients – the above-mentioned discreteness of the system and evolution equations which are nonlinear. As a result, a new paradigm of nonlinear science has recently emerged – the concept of discrete breathers (DB), equally labelled intrinsic localized modes (ILM) in solid state physics and discrete solitons (DS) in nonlinear optics. These exact solutions of a huge variety of underlying nonlinear lattice models are typically characterized by being time periodic and spatially localized, independent of the actual (assumed to be large) size of the lattice, independent of the spatial dimension of the lattice, mostly independent of the actual choice of nonlinear forces acting on the lattice, and so on. Mastering their mathematical properties in Hamiltonian lattices allows us to also include the effects of dissipation, dr
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