Completely integrable: Difference between revisions
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A few nonlinear dispersive and wave equations are lucky enough to be | 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. | 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. | This means in particular that they enjoy infinitely many conservation laws. | ||
In many cases these conservation laws provide control on high Sobolev | In many cases these conservation laws provide control on high Sobolev | ||
norms which seems to be a quite | 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 | allows to apply (at least formally) inverse scattering techniques to solve | ||
a completely integrable equation. Some nonlinear PDE's as the KdV posed | a completely integrable equation. Some nonlinear PDE's as the KdV posed | ||
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* [[Benjamin-Ono equation]] | * [[Benjamin-Ono equation]] | ||
* [[Cubic NLS|One dimensional cubic NLS]] | * [[Cubic NLS on R|One dimensional cubic NLS]] | ||
* [[Davey-Stewartson system]] | * [[Davey-Stewartson system]] | ||
* [[Kadomtsev-Petviashvili equation]] (both [[KP-I equation|KP-I]] and [[KP-II equation|KP-II]]) | * [[Kadomtsev-Petviashvili equation]] (both [[KP-I equation|KP-I]] and [[KP-II equation|KP-II]]) | ||
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* [[Sine-Gordon|Sine-Gordon equation]] | * [[Sine-Gordon|Sine-Gordon equation]] | ||
* [[wave maps on R|One-dimensional wave maps]] | * [[wave maps on R|One-dimensional wave maps]] | ||
[[Category:Integrability]] | [[Category:Integrability]] | ||
[[Category:Concept]] | [[Category:Concept]] | ||
Latest revision as of 08:40, 4 February 2011
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.
