View source for Comparison of solitons and waves

[[ru:Сравнение солитона с волной]]
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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 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+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 <span style="color:#808">purple outline</span>, the moving wave is shown by a <span style="background-color:#0bb">turquoise gradient</span>.

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Coded by [[Tsvetkov Denis]], the code written by Alexander Yavorsky ([https://github.com/yavalvas/kdf-equation/blob/master/kdf_equation.cpp link]) was used to accomplish the work.

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