Details

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].

Reference

[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.