Causal Interpretation of the Nonlinear Schrödinger Equation: An Analytic Example

The "self-focusing" nonlinear Schrödinger equation with , and so on, also known as the Gross–Pitaevskii equation, appears in wave propagation in nonlinear media such as signal transmission in optical fibers, Bose–Einstein condensation, and surface waves over sufficiently deep water. In addition, this model plays a major role in the theory of rogue waves. After the Korteweg–de Vries equation, this was the second nonlinear partial differential equation whose initial value problem was solved via the inverse scattering transform method. It has two types of solutions, one associated with a group of waves and the other in which the velocity depends on the amplitude. The first is a soliton whose envelope does not change in form over time. The second family of solutions are the so-called "breathers", which are rational solutions of the nonlinear Schrödinger equation.
In this Demonstration a spatially localized algebraic breather solution with only one oscillation is studied. The squared modulus of the wavefunction is interpreted as the particle density, where is a complex function and is its complex conjugate. The causal interpretation of quantum theory developed by Louis de Broglie and David Bohm 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. In the asymptotic case, the velocity of the particle becomes . In the graphic on the left, you can see the position of the particles, the squared wavefunction (blue), the quantum potential (red), and the velocity (green). In the center and on the right, the graphics show the squared wavefunction plus trajectories and the quantum potential plus trajectories.


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For the complex wavefunction, , where , , (initial width and amplitude) and (wavenumber) are real-valued constants. The analytic solution for the trajectories is taken from [1]. For , , and the wavefunction turns into the Peregrine soliton [2]. In the causal interpretation, only a few equations of motion are analytically integrable. In this Demonstration the equation of motion is derived by solving:
The analytical solution is given, and where is an integration constant. The initial position is estimated by solving the equation numerically. Real-valued trajectories are obtained for several 's. The path together with the starting point , lead to a trajectory in space, which is not the real configuration space.
[1] Z. Yan, "Financial Rogue Waves," arXiv:0911.4259v2 [q-fin.PR], 2010.
[2] D. H. Peregrine, "Water Waves, Nonlinear Schrödinger Equations and Their Solutions," Journal of the Australian Mathematical Society, Ser. B, 25, 1983 pp. 16–43.
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