Korteweg-de Vries equation

The Korteweg-de Vries (KdV) equation is

<math>u_t + u_xxx + 6uu_x = 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 [Korteweg-de Vries equation on R|on the line], [[Korteweg-de Vries equation on T|on the circle], and on the half-line.