One of the exact breather solutions

for

is a complex field as a function of

and

which is periodic in space

.

has several remarkable features (see [1]), including periodicity of the squared wavefunction with period

implicit analytic solutions of quantum trajectories for some values

a time-independent solution

for

and

, and a symmetry according to

, where

, with

. 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

is

, with the integration constant

.

On

YouTube 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.

[1] D. Schrader,

* **Asymptotisch Auftretende Solitonen-Lösungen der Nichtlinearen Schrödinger-Gleichung zu Beliebigen Secans-Hyperbolicus-Förmigen Anregungen*, Aachen, Germany: Shaker Verlag, 1998.