Soliton Trajectories of the Modified Korteweg-de Vries Equation (mKdV)
According to the Bohmian interpretation of quantum theory, often called today "causal de Broglie-Bohm theory", it is possible to allocate trajectories of idealized particles to waves. In this Demonstration the mKdV equation is studied to show how the velocity field guides the particles inside a two-soliton. From the mKdV, , the continuity equation is derived via , where is the velocity field and is the wave amplitude. As the value of the velocity is known at each time, the evolution of Bohmian trajectories is obtained by numerical integration for different initial positions. The two-soliton implies that the higher amplitude wave is narrower and faster than the wave with the minor amplitude. The trajectories of the individual solitons show that in the two-soliton collision the amplitude and the velocity are exchanged, rather than passing through one another. The system is time reversible: . On the left you can see the position of the particles, the wave amplitude (blue), and the velocity (red). On the right the graphic shows the wave amplitude and the complete trajectories in space. The velocity is scaled to fit.
The wave amplitude is taken from: A. Pekcan, "The Hirota Direct Method," master's thesis, Bilkent University, Turkey, 2005.