Difference between revisions of "Comparison of solitons and waves"

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[[ru:Сравнение солитона с волной]]
 
<!--[[ТМ|Кафедра ТМ]] > [[Проект "Термокристалл"]] > [[Compare soliton with wave]] <HR>
 
<!--[[ТМ|Кафедра ТМ]] > [[Проект "Термокристалл"]] > [[Compare soliton with wave]] <HR>
 
[[Виртуальная лаборатория]] > [[Compare soliton with wave]] <HR>-->
 
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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 the numerical solution we successively calculate derivatives <math>U'</math>, <math>U''</math> and <math>U'''</math>, using the central difference method:
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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>,

Latest revision as of 18:53, 18 January 2017


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 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.