Geometric Integrability of the Camassa–Holm Equation

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.

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