The Resonant Nonlinear Schrödinger Equation in the Causal Interpretation of Quantum Theory
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 of the wavefunction is interpreted as the particle number density, while the gradient of the phase 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 of quantum theory is a nonrelativistic theory of point particles moving along trajectories governed here by the RNLS equation. The trajectories are not directly detectable. In this Demonstration 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 .
On the right, the graphic shows the squared wavefunction and the trajectories. The left side shows the particle positions, the squared wavefunction (blue), the quantum potential (red), and the velocity (green). Check the “quantum potential" box to show the potential and the trajectories for a special case (, , , ).
Explicitly, the RNLS can be written: , with . The nonlinear potential , which could be considered a quantum potential, was first introduced by de Broglie and later by Bohm in a hidden-variable theory (later called a causal interpretation by Bohm) for quantum mechanics. This potential influences, in addition to the external force, the motion of the particles, independently of its intensity. Here, the quantum potential changes the dispersion of the soliton, because when becomes zero, the RNLS changes to the so-called defocusing NLS, admitting a "dark" soliton solution with nonvanishing boundary values. The quantum potential depends only on the form of the wave and in some cases does not fall off with distance.
By expressing the wavefunction in eikonal form with ( = particle number density), the RNLS equation reduces to two nonlinear partial differential equations, in which is the real-valued amplitude and is the real-valued phase of the wavefunction,
where the last term could be simplified, by multiplication of the above equation by , to the continuity equation
with velocity and particle density .
The comparison of the RNLS equation with the linear Schrödinger equation (LSE) shows that the nonlinear terms of the RNLS could be regarded as a special form of a potential in the LSE. In this case, only the gradient of the phase affects the velocity of the single quantum particle, because of the continuity equation for the particle density.
The analytical solution of the RNLS equation for a two-soliton is:
where , are the wave numbers and , are the initial positions of the maximum density. The system is time-reversible.
 O. K. Pashaev and J.-H. Lee, "Resonance Solitons as Black Holes in Madelung Fluid," Modern Physics Letters A 17(24), 2002 pp. 1601–1619. doi .10.1142/S0217732302007995.
 P. R. Holland, The Quantum Theory of Motion, Cambridge: Cambridge University Press, 1993.