**Eigenvector of ILM, The Shooting Method.**

Vary the mode frequency, **w**, by clicking the mouse on one of the
**red** or **blue** markers at the top of the
figure. The anharmonic mode eigenvector is constructed according to
the "rotating-wave" approximation (RWA). For some particular
eigenfrequency the mode eigenvector becomes localized. One may launch
a molecular-dynamics simulation (press MDS-button) and see the resulting
vibrational spectrum of the particles.

**It seems that this applet
does not work properly on Mac**

Within the rotating-wave approximation (RWA) [1] the eigenvector of the
anharmonic mode in the 1-D nonlinear chain has to satisfy the
following recurrence relations:
**
w**^{2} M f_{n} +
(f_{n+1} - 2 f_{n} + f_{n-1}) K_{2} +
(3/4) A^{2}
(f_{n+1} - f_{n})^{3} K_{4} +
(3/4) A^{2}
(f_{n-1} - f_{n})^{3} K_{4} = 0 ,
where **A** is the mode amplitude and **f**_{n}
represent the mode eigenvector.
In order to find the intrinsic local mode eigenvector we launch
"shots" [1] from the mode center and trying to form the localized shape for
the mode: when its amplitude decreases with distance away from the center.
By varying the mode frequency we can go through a variety of different
eigenvectors. At some particular eigenfrequency the mode eigenvector
becomes localized.

[1] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,
*Numerical Recipes in C*, (Cambridge Univ. Press, 1992) p. 749.

Last modified: August 13, 1997