An integrable version of the Resonant Nonlinear Schrödinger equation (RNLS) in 1+1 dimensions admits soliton solutions with a rich scattering structure. The RNLS equation can be interpreted as a particular realization of the nonlinear-Schrödinger-soliton propagating in a so-called "quantum potential". Recently, the RNLS was proposed to describe uniaxial waves in a cold collisionless plasma.
In quantum mechanics, the squared amplitude R of the wavefunction is interpreted as the particle number density, while the gradient of the phase S is proportional to the velocity of the quantum wave. The decomposition of the time-dependent linear Schrödinger wavefunction in configuration space into real and imaginary parts gives a pair of coupled nonlinear partial differential equations in which the real-valued amplitude and the real-valued phase of the wavefunction mutually determine one another. The real part is the Hamilton--Jacobi equation with an added term called the quantum potential; the imaginary part yields the continuity equation for the particle density. This method could be applied to the nonlinear version of the Schrödinger equation and especially to the RNLS equation.
The causal interpretation (often called: hidden variables theory, ontological interpretation, pilot wave theory or Bohmian mechanics) of quantum theory is a nonrelativistic theory of point particles moving along trajectories governed here by the RNLS equation. In this interpretation the wavefunction is a real wave in configuration space that imparts well-defined trajectories to quantum particles via quantum potential. In practice, it is impossible to predict or control the quantum trajectories with complete precision. Subsequently, these trajectories can be analyzed with the same methods as used with classical systems. In this video a two-soliton collision is studied. Inside the waves, the starting points for the particles are distributed according to the density of the waves at t=0.
On the right, the graphic shows the squared wavefunction and the trajectories. The left side shows the particle positions (yellow), the squared wavefunction (blue), the quantum potential (red), and the velocity (green)for the special case: (x1=5, x2=-5, p1=1.5, p2=2.75, time: -10≤t≤10). The velocity and quantum potential are scaled to fit. The system is time-reversible.
Programmed by: Klaus von Bloh
Bloh, Klaus von: "The Resonant Nonlinear Schrödinger Equation in the Causal Interpretation of Quantum Theory" from The Wolfram Demonstrations Project, http://demonstrations.wolfr...
..., June 2010.