# Bohm Trajectories for a Type of Derivative Nonlinear Schrödinger Equation

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The time-evolution of the quantum standard wavefunction is determined by the Schrödinger equation and guidance equation. The guidance equation states that the velocity field for the configuration is given by the quantum current divided by the density . The guidance equation is derived from the continuity equation, which is a special form of a conservation law.

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Contributed by: Klaus von Bloh (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The Hirota bilinear method is applied to find exact -soliton solutions for the DNLSE (see references). For this Demonstration a two-soliton solution is taken. The soliton has four free real parameters (, , , ) that characterize the velocity, the amplitude, and the width of the soliton. Due to the limited CPU power, only is chosen as a free parameter. The other parameters can be found in the para options in the program. The system is time-reversible.

The left graphic shows the position of the particles, the wave amplitude (blue), and the velocity (red). The right graphic shows the wave amplitude and the complete trajectories in (, ) space. The velocity is scaled to fit. In the program, if AccuracyGoal, PrecisionGoal, and MaxSteps are increased, the results will be more accurate.

References:

J. Lee, Y. Lee, and C. Lin, "Exact Solutions of DNLS and Derivative Reaction-Diffusion Systems," *Journal of Nonlinear Mathematical Physics*, 9(1), 2002 pp. 87–97.

W. Struyve and A. Valentini, "De Broglie-Bohm Guidance Equations for Arbitrary Hamiltonians," ArXiv:quant-ph/0808.0290v2, 2008 pp. 1–22.

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