Eigenvector of ILM, The Shooting Method.
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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.
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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" [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