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Time Evolution of Optical Rogue Waves (Rogons) in the Causal Interpretation of Quantum Mechanics

In this Demonstration, modified Peregrine solitons resembling optical rogue waves (here called rogons) are studied in the causal interpretation of L. de Broglie and D. Bohm. The rogons have a variable function and a variable initial position (for the first case). The word rogon has its origin in physical similarities between the rogue waves and solitary wave. If the waves reappear virtually unaffected in size or shape shortly after their interactions, Yan [1, 3] called them "rogons".
The causal interpretation (also called hidden variables theory, ontological interpretation, de Broglie–Bohm theory, pilot-wave theory or Bohmian mechanics) has the feature that particles are regarded as both particles and waves, where the wave in the configuration space affects the motion of the particle. The trajectories could be interpreted as streamlines of a certain kind of fluid. The complex-valued wave obeys the nonlinear Schrödinger equation and the guiding equation (velocity) of the particles is derived by the continuity equation.
The graphic on the left shows the density (blue), the quantum potential (red), and the velocity (green). On the right, you can see the density and the trajectories in -space. The velocity and the quantum potential are scaled to fit.

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For simplicity, the homogeneous nonlinear Schrödinger equation with only one variable coefficient is taken for the evaluation of the de Broglie–Bohm trajectories [1]:
,
with and so on, where is a complex function and is its complex conjugate.
The causal interpretation introduced trajectories that are guided by a real phase function determined by the wavefunction. In the eikonal representation of the wave, , the gradient of the phase is the particle velocity . In this interpretation, the origin of the motion of the particle is the potential , given by , plus an additional term . The latter may be interpreted alternatively as a kind of fluid pressure or quantum Bohm potential.
For this Demonstration the three functions and are
(1) and ,
(2) and ,
(3) and ,
where is a free parameter and is the initial position of the peak of the wave. For the first two cases there are analytical solutions for the trajectories; let be an integration constant. The initial positions are estimated by solving the equation numerically. Real-valued trajectories are obtained for several 's. The starting positions of the particles are linearly distributed around the peak of the wave at and ( only for the first case). Due to insufficient decrease in the merit function of the integration constant in the analytic functions of the trajectories, only the positive initial positions (right from the peak) are evaluated, through Mathematica's built-in function FindRoot. The trajectories for the negative initial positions (left from the peak) are computed numerically by Mathematica's built-in function NDSolve. You can control the paths together with the starting points by changing the initial condition . The path together with the starting point leads to a trajectory in -space.
References
[1] Z. Yan, "Nonautonomous 'Rogons' in the Inhomogeneous Nonlinear Schrödinger Equation with Variable Coefficients," Physics Letters A, 374, 2010 pp. 672–679.
[2] W. Xiao-Chun, H. Jing-Song, and L. Yi-Shen, "Rogue Wave with a Controllable Center of Nonlinear Schrödinger Equation," Communications in Theoretical Physics, 56, 2011 pp. 631–637.
[3] Z. Yan. "Optical Rogue Waves (Rogons)." The Wolfram Demonstrations Project. demonstrations.wolfram.com/OpticalRogueWavesRogons.
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