Completely integrable

A few nonlinear dispersive and wave equations are lucky enough to be completely integrable in the sense that there exist a Lax pair formulation of the equation. This means in particular that they enjoy infinitely many conservation laws. In many cases these conservation laws provide control on high Sobolev norms which seems to be a quite exceptional event. The existence of a Lax pair allows to apply (at least formally) inverse scattering techniques to solve a completely integrable equation. Some nonlinear PDE's as the KdV posed on the circle are integrable in the sense of Liouville-Arnold which means that there exist action/angle variables associated to the equation.

List of completely integrable nonlinear dispersive and wave equations

• Benjamin-Ono equation
• One dimensional cubic NLS
• Davey-Stewartson system
• Kadomtsev-Petviashvili equation (both KP-I and KP-II)
• Korteweg-de Vries equation, and more generally the KdV hierarchy
• Modified Korteweg-de Vries equation
• Sine-Gordon equation
• One-dimensional wave maps