Two soliton solution of the KdV equation

The KdV equation

u_t = - u_xxx - 6uu_x

describes long water waves in shallow water.

Among many remarkable properties (all stemming from the fact that it is "integrable" in some sense), the nonlinear term (the second term on the right hand side) somehow exactly compensates for the dispersive term (the first term on the right hand side). This allows it to admit "soliton" solutions.

Furthermore, higher solitons travel faster. This means a tall soliton will catch up with a short one, and they will undergo some strange interaction. Amazingly, despite the nonlinearity, a long time afterwards they seem to recover their original forms (apart from a phase shift). In other words, the tall one seems to have overtaken the short one!

This preservation of structure suggests that there are many quantities conserved under evolution, and in fact, there are infinitely many.

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Tags:

• KdV
• solitons

[kubaaa222]

these are amazing!

[anayugu]

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