fig1.GIF
Fig. 1a shows the wavelet transform of breather of amplitude $A'= .30$
and frequency $\omega_b = 2.96$ Hz. A
harmonic can be seen at $2 \times \omega_b =5.92Hz$. Note that the vertical
scale is logarithmic. The phonon band is bracketted by grey lines.
Fig. 1b: Increasing the amplitude
to $A'= .32$ lowers the frequency to 2.93 Hz and the
harmonic above the band shifts towards the linear band.
fig2.GIF
Fig. 2a shows the frequencies present in
a breather of amplitude $A'= .34$. Secondary
frequencies appear slightly above and below the main breather
frequency (at $\omega_b = 2.93$) in addition to the harmonic at $2 \times \omega_b$.
Fig. 2b shows the frequencies which result from the collapse of
a breather of amplitude $A'= .342$. The collapse itself is quite sudden and is
not adequately captured by the wavelet transform. Note the very faint frequency
bands visible slightly above 6Hz and in the phonon band.
fig3.GIF
Fig. 3 shows the transform of an initial
sine wave of cycle number $p=160$ which falls in the
region of stability.
fig4.GIF
Fig. 4a and Fig. 5a show the wavelet transform of the formation of breather
modes from the bottom
edge of the linear band for a initial sine wave of
p = 14 and 16 cycles, respectively. Figures 4b and 5b show the related
amplitude and the combined energy of the three sites centered on the
breather for the same time period. The amplitude is shown by the shaded
region and the amplitude is indicated with a dark line.
fig6.GIF
Fig. 6 shows the length of time until the amplitude is greather than twice
the initial amplitude somewhere on the chain
for initial sine waves of different wave vectors $q = 2 p \pi/N$
where p, the number of cycles on the chain, is an integer. The
boxes indicate waves modulated with a sine wave of wavelength 48,
the + symbol indicates modulation with a sine wave of
wavelength 24 and the circles indicate a random modulation. All modulations
were of maximum magnitude 0.01 compared to the initial sine wave condition of
amplitude 0.14.
fig7.GIF
Fig. 7a shows the wavelet transform of the case of an
initial traveling sine wave with 40 cycles on the chain of 600 sites.
Fig. 7b shows the amplitude (shaded) of the site and it shows the energy
(dark line) of the three sites
centered on the maximum energy site. As can be
seen from the transform, as the wave attempts to grow in size additional
frequencies appear near the top of the band which prevent further growth.
fig8.GIF
Fig. 8a and fig. 8b show respectively the actual amplitude and the spatial wavelet transform
of the entire chain of 600 sites at an early time slice for 6 cycles on the
chain. Fig. 8c and fig. 8d show respectively
the actual amplitude and the spatial wavelet transform of the
entire chain of 600 sites at a later time slice for the same initial conditions.
Clearly the 6 Hz signal is disappearing and is being replaced with Dirac delta
type spikes which represent multiple spatial frequencies needed to describe
a very narrow discrete breather.
Quick Time movie
QuickTime movie of the evolution of Fig. 8a to Fig. 8b. (QuickTime movie; 204k)
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Contact Kyle Forinash for comments/suggestions/corrections.