${\displaystyle i\varphi _{t}+\varphi _{xx}=0,}$

${\displaystyle i^{2}=-1}$

${\displaystyle x}$

${\displaystyle t}$

The wave-packet solution applied here is:

${\displaystyle \varphi (x,t)=a\,{\frac {{\text{e}}^{-{\tfrac {1}{8}}v^{2}}}{\sqrt {1+2it}}}\,\exp \left(-{\frac {1}{2}}{\frac {(x-{\tfrac {1}{2}}iv)^{2}}{1+2it}}\right),}$

with ${\displaystyle v}$ the group velocity and ${\displaystyle a}$ the amplitude. This solution has an envelope of Gaussian shape:

${\displaystyle |\varphi (x,t)|={\frac {|a|}{\sqrt[{4}]{1+4t^{2}}}}\exp \left(-{\frac {1}{2}}\,{\frac {(x-vt)^{2}}{1+4t^{2}}}\right).}$

So the envelope of the linear solution ${\displaystyle {|\varphi (x,t)|}}$ widens with time, and becomes lower. The centre of gravity ${\displaystyle x_{c}(t)}$ of the wave packet is computed as ${\displaystyle \textstyle x_{c}=\int x\,|\varphi (x,t)|^{2}\,{\text{d}}x/\int |\varphi (x,t)|^{2}\,{\text{d}}x.}$

${\displaystyle a=2}$

${\displaystyle v=4.}$

${\displaystyle \omega =k^{2}}$

${\displaystyle \omega }$

${\displaystyle k}$