Details

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.