While it is not surprising that the loss of periodicity in defect crystals leads to localized vibrational phenomena, the standard description of the dynamics of defect-free periodic lattices in terms of plane wave phonons is so deeply ingrained that it was surprising to many researchers when it was argued theoretically in the latter half of the 1980's [1-3] that the presence of strong quartic anharmonicity in perfect lattices can also lead to localized vibrational modes, henceforth called "intrinsic localized modes" (ILMs). Sievers and Takeno studied the classical vibrational dynamics of a simple one-dimensional monatomic chain of particles interacting via nearest-neighbor harmonic and quartic anharmonic springs, and they used a "rotating wave approximation" (RWA), in which just one frequency component was kept in the time dependence. For the case of sufficiently strong positive quartic anharmonicity, it was found that the lattice could sustain stationary localized vibrations having the approximate mode pattern A(...,0,-1/2,1,-1/2,0,...), where A is the amplitude of the "central" atom. This pattern is odd under reflection in the central site and is "optic mode" -like, in that adjacent particles move p out of phase. As with all nonlinear vibrations, the ILM frequencies are amplitude dependent. The ILMs can be centered on any lattice site, giving rise to a configurational entropy analogous to that for vacancies, and some thermodynamic ramifications were explored along with speculations about the possible presence of ILMs in strongly anharmonic solids such as solid He and ferroelectrics[3,4].

In the years since the appearance of these papers, there has been a rapidly increasing number of theoretical studies published related to ILMs. The ILM papers published since 1988 roughly divide into two broad and overlapping categories. The first of these involves the relation of the new excitations to the general behavior of discrete nonlinear systems, with emphasis on connections to soliton-like behavior in lattices. The second category is more tightly focused on generalizations and extensions of the 1988 papers, in order to discover the properties of highly-localized large-amplitude ILMs and also to assess their likely importance for real solids. Computer studies of the dynamics of nonlinear lattices extend back to the pioneering work on one-dimensional chains by Pasta, Fermi and Ulam [5] , and they have been a vital adjunct to much of the subsequent work on solitons in anharmonic lattices. Discussions of both analytic and numerical aspects of soliton behavior in one-dimensional lattices with intersite cubic and quartic anharmonic interactions are found in Flytzanis, et al. [6] for the monatomic case and in Pnevmatikos, et al. [7] for the diatomic case. Of particular interest for ILMs is their relationship to lattice envelope solitons. Stationary lattice solitons were studied theoretically over 20 years ago by Kosevich and Kovalev [8] for anharmonic chains with cubic and quartic onsite and intersite anharmonicity. The emphasis was on onsite anharmonicity, and solutions were obtained for low-anharmonicity stationary envelope solitons whose spatial extent is broad compared with the lattice constant. Later MD simulations by Yoshimura and Watanabe [9] for linear chains with quadratic plus quartic interactions showed that the stationary envelope soliton solutions account well for ILMs whose spatial widths are sufficiently broad, e.g. 10 or more lattice constants. However, as was subsequently emphasized by Kosevich [10], the stationary envelope soliton solutions do not describe the large-anharmonicity highly localized ILMs which are stabilized by lattice discreteness.

This history is after Ref. [11] ...

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