The nonlinear Schrödinger equation is

, with

, and so on, where

is a complex function and

is its complex conjugate, and the squared modulus

of the wavefunction is interpreted as the wave density. Here standard notation is used, in which

is the temporal and

is the spatial coordinate. An analytic solution for the quantum motion can be found. The wavefunction with an arbitrary linear time-dependent potential

becomes

.

Here

is a one-soliton solution to the nonlinear Schrödinger equation from which the velocity

for the single particles, guided by the wave density obeying the continuity equation, is expressed by

, where the path

of an individual particle is derived by

, where

is the starting point of the path for

. If

exists and, for the first two cases, with

, there is an analytical solution of the motion. In the causal interpretation, the origin of the motion for the nonlinear Schrödinger equation of the particle is the effective potential

, given by

, plus an additional term

, which is called quantum potential. The starting positions of the particles are linearly distributed around the peak of the wave, which depends on the parameter

. The path

together with the starting point

leads to a trajectory in

space. The time-dependent functions

are

.

You can control the paths together with the starting points by changing the initial conditions

The velocity, acceleration, and quantum potential are appropriately scaled to fit.

The solution is taken from [1].

[1] Z.-D. Li, Q.-Y. Li, X.-H. Hu, Z.-X. Zheng, and Y.-B. Sun, "Hirota Method for the Nonlinear Schrödinger Equation with an Arbitrary Linear Time-Dependent Potential,"

* Annals of Physics*,

** 322**(11), 2007 pp. 2545–2553.