A Breather Solution in the Causal Interpretation of Quantum Mechanics
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A breather solution appears often in nonlinear wave mechanics, in which a nonlinear wave has energy concentrated in a localized oscillatory manner. This Demonstration studies a breather solution with a hyperbolic secant envelope of the focusing nonlinear Schrödinger (NLS) equation , with and so on, also known as the Gross–Pitaevskii equation in the causal interpretation, developed by Louis de Broglie and David Bohm.[more]
The NLS equation could be interpreted as the Schrödinger equation (SE) with the nonlinear potential term with , although for most situations it has no relationship with the quantum Schrödinger equation other than in name. In the studied breather, there are large density amplitudes at certain times, which could be interpreted as rogue waves.
The graphic on the left shows the density (blue), the quantum potential (red), and the velocity (green). On the middle and on the right, you can see the density and the trajectories in space and the quantum potential and the trajectories in space. The velocity and the quantum potential on the left side are scaled to fit.[less]
Contributed by: Klaus von Bloh (July 2014)
Open content licensed under CC BY-NC-SA
One of the exact breather solutions for is a complex field as a function of and which is periodic in space .
The complex-valued wavefunction
, with ,
has several remarkable features (see ), including periodicity of the squared wavefunction with period implicit analytic solutions of quantum trajectories for some values a time-independent solution for and , and a symmetry according to , where , with . For the case , the wavefunction becomes , where an implicit analytic solution for the quantum motion is given by the gradient of the phase function of the wavefunction in the eikonal form (often called polar form). For this special case, the implicit function for is , with the integration constant .
Exploiting to the symmetry of the wave density, only one half of the trajectories were calculated numerically.
On YouTube there are some videos by the author, which show additional breather solutions with a hyperbolic secant envelope in the de Broglie–Bohm interpretation for the Gross–Pitavevskii equation.
 D. Schrader, Asymptotisch Auftretende Solitonen-Lösungen der Nichtlinearen Schrödinger-Gleichung zu Beliebigen Secans-Hyperbolicus-Förmigen Anregungen, Aachen, Germany: Shaker Verlag, 1998.
 Bohmian-Mechanics.net. (Jul 11, 2014) www.bohmian-mechanics.net/index.html.
 S. Goldstein. "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy. (Jul 11, 2014)plato.stanford.edu/entries/qm-bohm.