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

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" [2] 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.

Eigenvector of a train of anharmonic local modes versus
eigenfrequency.
All modes have
the same amplitude
(**A**=0.1) with their normalized frequencies shown at the
right-top corner of the each panel:
(a) w_{loc}/w_{m}=1.24723,
(b) w_{loc}/w_{m}=1.24829,
(c) w_{loc}/w_{m}=1.24830.
The eigenvectors are for a monatomic chain of particles of
mass **M**=1,
interacting through the nearest-neighbor potential with
**K**_{2}=1,
**K**_{4}=10.

[1] A.J.Sievers and S.Takeno, Phys. Rev. Lett. **61**, 970 (1988).

[2] 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 12, 1997