Korteweg-de Vries equation

The Korteweg-de Vries (KdV) equation is

$\partial _{t}u+\partial _{x}^{3}u+6u\partial _{x}u=0.$

The factor of 6 is convenient for reasons of complete integrability, but can easily be scaled out if desired.

The equation is completely integrable, and has infinitely many conserved quantities. Indeed, for each non-negative integer k, there is a conserved quantity which is roughly equivalent to the $H^{k}$ norm of u.

The KdV equation has been studied on the line, on the circle, and on the half-line.

The KdV equation is the first non-trivial equation on the KdV hierarchy and is the most famous member of the family of KdV-type equations.