# Soliton Trajectories of the Modified Korteweg-de Vries Equation (mKdV)

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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.

Contributed by: Klaus von Bloh (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The wave amplitude is taken from: A. Pekcan, "The Hirota Direct Method," master's thesis, Bilkent University, Turkey, 2005.