Consider a chain of particles of mass

Such lattice supports intrinsic local modes (ILMs) with their
frequencies above the phonon band characterized by the maximal
harmonic plane waves frequency
.

The eigenvector of the intrinsic local mode can be found within the
rotating-wave apporximation (RWA) where the displacement of the
*n*th particle from its equilibrium position
** u_{n}**
is described by the following ansatz

where is the amplitude of the mode, and characterizes its ac displacement pattern. Substitution of the above ansatz into the classical equations of motion

allows one to find the mode eigenvector. The ILM's eigenvector is a wave package which transfers to a lattice envelope soliton in a limit of a weak anharmonicicty.

A similar ansatz can give the eigenvector of a moving ILM.

A more complete description of the intrinsic local modes you can find in
a review article:

S. A. Kiselev, S. R. Bickham, and A. J. Sievers,
"Properties of Intrinsic Localized Modes in One-Dimensional Lattices",
*Comments Cond. Mat. Phys*, **17**, 135-173 (1995).

The above

You can launch either an

The time is shown in units of the shortest period of small amplitude plane wave vibrations, .

Energy is shown in arbitrary units. The **kinetic**
energy of the particle and the **potential**
energy of the bond are shown as the **red** and the
**yellow** bars, respectively.

If you wait for a while you will see a spectrum of the particles' vibrations. It will be shown in the left panel. As the time of the evolution goes the spectrum resolution improves. The frequency unit is the maximal plane wave frequency, .

Last modified: December 1, 1996

Sergey Kiselev,