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:
w2 M fn +
(fn+1 - 2 fn + fn-1) K2 +
(3/4) A2
(fn+1 - fn)3 K4 +
(3/4) A2
(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" [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) wloc/wm=1.24723,
(b) wloc/wm=1.24829,
(c) wloc/wm=1.24830.
The eigenvectors are for a monatomic chain of particles of
mass M=1,
interacting through the nearest-neighbor potential with
K2=1,
K4=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