Soliton Trajectories According to the Pilot Wave Theory

This video shows a particle-based fluid trajectory description of two-soliton interactions. The non-linear Korteweg-de Vries equation (KdV) supports multi-soliton-solutions. These solutions produce a pattern of N straight lines in the (x,t)-plane. In particular, the KdV equation describes approximately the slow evolution of water waves in shallow water of uniform depth. Solitons are stable waves in space and time, in which the velocity depends on the amplitude. When solitons interact with other solitons, their shapes do not change, but their phase shifts. The trajectory method (called pilot wave theory or causal interpretation) for quantum motion, developed by Louis de Broglie and David Bohm, is applied to the KdV equation. In this method the motion of idealized particles is governed by the current flow, which is derived directly from the continuity equation; u(x,t) is proportional to the density of the wave in which idealized particles are positioned at the point x at time t. The current flow divided by the density establishes the guiding equation (velocity field) for the individual path of the particles in the wave. On the left you can see the position of the particles (white), the wave amplitude (yellow) and the velocity (red). On the right the graphic shows the squared wavefunction and the complete trajectories. The velocity is scaled to fit. The particle positions are plotted against their downscaled kinetic energy along the vertical axis.

Programmed by: Klaus von Bloh

References:
[1] Bloh, Klaus von, "Soliton Trajectories According to Bohmian Quantum Mechanics" from The Wolfram Demonstrations Project, http://demonstrations.wolfram.com/SolitonTrajectoriesAccordingToBohmianQuantu..., 2008.
[2] R. Hirota, "The Direct Method in Soliton Theory", Cambridge, UK: Cambridge University Press, 2004.
[3] P. Holland, "The Quantum Theory of Motion", Cambridge, UK: Cambridge University Press, 1993.

Category:

Education

Tags:

• pilot wave theory
• Computer simulation
• ontological interpretation
• causal interpretation
• David Bohm
• Bohmian mechanics
• dBB
• quantum motion