One-Soliton Nonlinear Schrödinger Equation with Arbitrary Linear Time-Dependent Potential
This Demonstration considers the nonlinear Schrödinger equation, also known as the Gross–Pitaevskii equation. This pertains to an arbitrary linear time-dependent potential in the causal interpretation for a bright one-soliton solution, equivalent to the dynamics of a soliton solution in one-dimensional Bose–Einstein condensate. The existence of soliton solutions is a general feature of nonlinear wave equations, to which this trajectory concept is applied.[more]
The causal interpretation of quantum theory developed by Louis de Broglie and David Bohm introduced trajectories that are guided by a quantum potential. The velocities of single particles, following the trajectories, are determined by the phase of the wavefunction. The system is time reversible.
Despite the problems of the trajectory concept in quantum theory, the causal interpretation introduces a particle-based theory in the nonlinear wave mechanics naturally. The time evolution of the position could be interpreted as streamlines of the wave or as idealized test particles that do not interact on each other and do not perturb the wave.
In the graphic on the left, you can see the position of the particles, the squared wavefunction (black), the effective potential term (blue), the quantum potential (red), and the velocity (green). On the right, the graphic shows the squared wavefunction, as well as the trajectories.[less]
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 .
 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.