One of the exact breather solutions
is a complex field as a function of
which is periodic in space
has several remarkable features (see ), including periodicity of the squared wavefunction with period
implicit analytic solutions of quantum trajectories for some values
a time-independent solution
, and a symmetry according to
. For the case
, the wavefunction becomes
, where an implicit analytic solution for the quantum motion
is given by the gradient of the phase function of the wavefunction in the eikonal form (often called polar form). For this special case, the implicit function for
, with the integration constant
there are some videos by the author
, which show additional breather solutions with a hyperbolic secant envelope in the de Broglie–Bohm interpretation for the Gross–Pitavevskii equation.
 D. Schrader, Asymptotisch Auftretende Solitonen-Lösungen der Nichtlinearen Schrödinger-Gleichung zu Beliebigen Secans-Hyperbolicus-Förmigen Anregungen
, Aachen, Germany: Shaker Verlag, 1998.