One 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 that preserve their structure under time evolution.

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

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• KdV
• solitons

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