Comparison of solitons and waves

To demonstrate a soliton propagation, we numerically solve the Korteweg–de Vries equation. It has the form:

[math]\dot U + UU' + CU''' = 0[/math].

To obtain the numerical solution we successively calculate derivatives [math]U'[/math], [math]U''[/math] and [math]U'''[/math], using the central difference method:

[math]U' = \frac{U_{n+1} - U_{n-1}}{2\Delta x}[/math],

[math]U'' = \frac{U_{n+2} - 2U_{n} + U_{n-2}}{4\Delta x} \approx \frac{U_{n+1} - 2U_{n} + U_{n-1}}{2\Delta x}[/math],

[math]U''' = \frac{U_{n+2} - 2U_{n+1} + 2U_{n-1} - U_{n-2}}{4\Delta x}[/math].

The soliton is indicated by the purple outline, the moving wave is shown by a turquoise gradient.

Coded by Tsvetkov Denis, the code written by Alexander Yavorsky (link) was used to accomplish the work.