Ruggero Vaia - ISC-CNR Firenze #
Dispersion of a Traveling Wavepacket #
In several problems concerning 1D dynamics, e.g., quantum-state 
transmission, one is faced with the dispersive evolution of an 
input wavepacket, whose Fourier-space components are determined 
by its initial shape. The evolution occurs along a `wire' and is 
substantially ruled by its dispersion relation, which is usually 
a nonlinear function of the (quasi-) momentum when the wire is 
realized by discrete arrays of physical objects. It is textbook 
knowledge that a Gaussian packet broadens with a rate depending 
on the second derivative of the dispersion relation.
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In order to preserve as much as possible the wavepacket shape 
one must avoid dispersion: it is therefore convenient to 
initialize the wavepacket with its components sitting aroung an 
inflection point of the dispersion relation, so that 
higher-order terms determine the dispersion.
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In the literature the role of the cubic nonlinearity of is 
accounted for in the case of a Gaussian packet. However, there 
are reasons to look for an extension of this
result: besides the possibility that cubic terms could also 
vanish (e.g., by symmetry), one could be interested in a 
non-Gaussian initial shape of the wavepacket. In this work such 
an extension is obtained in terms of rather simple formulas. 
These permit to obtain an optimal initial width, which shows 
peculiar scaling as a function of the wire length.