On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations

Abstract

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.

arXiv comment26 pages

References in corpus (4)

• An integrable shallow water equation with peaked solitons
• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• The Camassa-Holm equation as a geodesic flow on the diffeomorphism group
• Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms

Cited by in corpus (3)

• Global existence and blow-up for a weakly dissipative $\mu$DP equation
• On the Cauchy problem of a periodic 2-component $\mu$-Hunter-Saxton equation
• On the Cauchy problem of a weakly dissipative $\mu$HS equation