Eigenvector of ILM, The Shooting Method.
Within the rotating-wave approximation (RWA)  the eigenvector of the
anharmonic mode in the 1-D nonlinear chain has to satisfy the
following recurrence relations:
w2 M fn +
(fn+1 - 2 fn + fn-1) K2 +
(fn+1 - fn)3 K4 +
(fn-1 - fn)3 K4 = 0 ,
where A is the mode amplitude and fn
represent the mode eigenvector.
In order to find the intrinsic local mode eigenvector we launch
"shots"  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
Eigenvector of a train of anharmonic local modes versus
All modes have
the same amplitude
(A=0.1) with their normalized frequencies shown at the
right-top corner of the each panel:
The eigenvectors are for a monatomic chain of particles of
interacting through the nearest-neighbor potential with
 A.J.Sievers and S.Takeno, Phys. Rev. Lett. 61, 970 (1988).
 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