Letters in Mathematical Physics

, Volume 59, Issue 2, pp 117-131

First online:

Geometric Integrability of the Camassa–Holm Equation

  • Enrique G. ReyesAffiliated withDepartment of Mathematics, Yale University

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


It is observed that the Camassa–Holm equation describes pseudo-spherical surfaces and that therefore, its integrability properties can be studied by geometrical means. An sl(2, R)-valued linear problem whose integrability condition is the Camassa–Holm equation is presented, a ‘Miura transform’ and a ‘modified Camassa–Holm equation’ are introduced, and conservation laws for the Camassa–Holm equation are then directly constructed. Finally, it is pointed out that this equation possesses a nonlocal symmetry, and its flow is explicitly computed.

Camassa–Holm equation pseudo-spherical surfaces geometric integrability Miura transformation conservation law nonlocal symmetry