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.