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On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations. (English) Zbl 1234.35222
Summary: Considered herein are the generalized Camassa-Holm and Degasperis-Procesi equations in the spatially periodic setting. The precise blow-up scenarios of strong solutions are derived for both of equations. Several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established. The exact blow-up rates are also determined. Finally, geometric descriptions of these two integrable equations from non-stretching invariant curve flows in centro-equiaffine geometries, pseudo-spherical surfaces and affine surfaces are given.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35Q35 PDEs in connection with fluid mechanics
35B44 Blow-up in context of PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35D35 Strong solutions to PDEs
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