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Cell["Conclusions and questions", "Section"],
Cell[CellGroupData[{
Cell["Length Scale", "Subsection"],
Cell["\<\
An alternative idea to consider for the width of a soliton-like pulse is at \
what length scale a smooth potential will have the most interaction with it. \
For periodic potentials as I have investigated here, I would consider this to \
be the period of the potential which causes the maximum disruption of an \
initial pulse. \
\>", "Text"],
Cell[TextData[{
" WWith the initial pulses used here, and the potential -5 Cos[\[Omega] x], \
this was at \[Omega] = 4, or a period of \[Pi]/2. This is different, but of \
the same order of magnitude with other measures, such as the variance, which \
is ",
Cell[BoxData[
\(TraditionalForm\`\[Pi]\^2/4\)]],
" (~2.47), or the width at half height, which is ",
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" (~2.63)."
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"However, the thing to keep in mind is that this is a ",
StyleBox["nonlinear ",
FontWeight->"Bold"],
"measure. It depends on the size of the potential -- with the potential \
-3/5 Cos[\[Omega] x], the maximum is at about \[Omega] = 1.5, or a period of \
about 4.2. Perhaps the proper length scale to consider for a soliton0like \
pulse is so hard to pin down precisely because it is a nonlinear dynamic \
quantity."
}], "Text"],
Cell["\<\
In any case, it is clear that across the range of frequencies I have shown, \
the behavior of the pulse has varied from particle-like where it bounces back \
and forth in the potential well to group-like where the pulse as a whole \
appears to average out the effects of the potential.\
\>", "Text"]
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Cell[CellGroupData[{
Cell["Questions", "Subsection"],
Cell["\<\
This study seems to have progressed to the point where I have produced more \
questions than I started with. Here are a few I would like to find answers \
for.\
\>", "Text"],
Cell["\<\
Will the next order in the slowly varying potential approximation be \
sufficient to give cleaner results, and perhaps more importantly from the \
above conclusions give an indication as to when it breaks down?\
\>", "Text"],
Cell["\<\
Can I find an approximate theory for a rapidly varying potential which does \
nearly as well as the slowly varying potential? \
\>", "Text"],
Cell["\<\
Finally, is there a reasonable way to relate these length scale results to \
the effect of a random potential?\
\>", "Text"]
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