The Regular Benjamin-Ono Equation is a Soliton Equation. It has 1-Soliton, 2-Soliton , and in general N-Soliton Solutions.

One-Soliton Solution

The One Soliton Solution of the B-O equation is given by a Lorentzian Profile.

This corresponds to the following:

An animation of the above follows below (each time frame is plotted in a separate frame, but shown in rapid succession)

Multi-Soliton Solution

The N-Soliton Solutions of the Benjamin-Ono Equation are given by a series of assignments and formulas.

One-Periodic Soliton Solution

The Single Soliton Solution for the Periodic Conditions is given by:

See the animation of the evolution of the above soliton here:

Multi-Periodic Soliton Solution

Then we define several new quantities based on these given parameters as follows:

Finally, we have that :

The animation of the evolution of this soliton is as follows:

Created by Mathematica (April 26, 2006) |