Two-Soliton Collision for the Gross-Pitaevskii Equation in the Causal Interpretation

Under certain simplified assumptions (small amplitudes, propagation in one direction, etc.), various dynamical equations can be solved, for example, the well-known nonlinear Schrödinger equation (NLS), also known as the Gross–Pitaevskii equation. It has a soliton solution, whose envelope does not change in form over time. Soliton waves have been observed in optical fibers, optical solitons being caused by a cancellation of nonlinear and dispersive effects in the medium. When solitons interact with one another, their shapes do not change, but their phases shift. The two-soliton collision shows that the interaction peak is always greater than the sum of the individual soliton amplitudes. The causal interpretation of quantum theory is a nonrelativistic theory picturing point particles moving along trajectories, here, governed by the nonlinear Schrödinger equation. It provides a deterministic description of quantum motion by assuming that besides the classical forces, an additional quantum potential acts on the particle and leads to a time-dependent quantum force . When the quantum potential in the effective potential is negligible, the equation for the force will reduce to the standard Newtonian equations of classical mechanics. In the two-soliton case, only two of the Bohmian trajectories correspond to reality; all the others represent possible alternative paths depending on the initial configuration. The trajectories of the individual solitons show that in the two-soliton collision, amplitude and velocity are exchanged, rather than passing through one another. On the left you can see the position of the particles, the wave amplitude (blue), and the velocity (green). On the right the graphic shows the wave amplitude and the complete trajectories in (, ) space.


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With the potential , the Gross–Pitaevskii equation is the nonlinear version of the Schrödinger equation , where is the complex conjugate and the density of the wavefunction.
The exact two-soliton solution is:
, with
and , and here .
There are two ways to derive the velocity equation: (1) directly from the continuity equation, where the motion of the particle is governed by the current flow; and (2) from the eikonal representation of the wave, , where the gradient of the phase is the particle velocity. Therefore, the quantum wave guides the particles. The origin of the motion of the quantum particle is the effective potential , which is the quantum potential plus the potential , . The effective potential is a generalization of the quantum potential in the case of the Schrödinger equation for a free quantum particle, where . The system is time reversible. In the source code the quantum potential is deactivated, because of the excessive computation time.
J. P. Gordon, "Interaction Forces among Solitons in Optical Fibers", Optics Letters, 8(11), 1983 pp. 596–598.
P. Holland, The Quantum Theory of Motion, Cambridge: Cambridge University Press, 1993.
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