Difference between revisions of "Comparison of solitons and waves"
From Department of Theoretical and Applied Mechanics
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To demonstrate a soliton propagation, we numerically solve the Korteweg–de Vries equation. It has the form: | To demonstrate a soliton propagation, we numerically solve the Korteweg–de Vries equation. It has the form: | ||
::<math>\dot U + UU' + CU''' = 0</math>. | ::<math>\dot U + UU' + CU''' = 0</math>. | ||
| − | To obtain | + | To obtain a 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+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} + U_{n-2}}{4\Delta x} \approx \frac{U_{n+1} - 2U_{n} + U_{n-1}}{2\Delta x}</math>, | ||
Revision as of 02:31, 9 June 2016
To demonstrate a soliton propagation, we numerically solve the Korteweg–de Vries equation. It has the form:
- .
To obtain a numerical solution we successively calculate derivatives , and , using the central difference method:
- ,
- ,
- .
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.
