Modified Korteweg-de Vries equation
The (defocusing) modified Korteweg-de Vries (mKdV) equation is
It 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. This equation has been studied on the line, on the circle, and on the half-line.
The focussing mKdV
is very similar, but admits soliton solutions.