A continuous interpolation between conservative and dissipative solutions for the two-component Camassa-Holm system

Abstract

We introduce a novel solution concept, denoted $\alpha$-dissipative solutions, that provides a continuous interpolation between conservative and dissipative solutions of the Cauchy problem for the two-component Camassa-Holm system on the line with vanishing asymptotics. All the $\alpha$-dissipative solutions are global weak solutions of the same equation in Eulerian coordinates, yet they exhibit rather distinct behavior at wave breaking. The solutions are constructed after a transformation into Lagrangian variables, where the solution is carefully modified at wave breaking.

58 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1301.1445

References in corpus (2)

• On an integrable two-component Camassa-Holm shallow water system
• Singular solutions of a modified two-component Camassa-Holm equation

Cited by in corpus (2)

• Real-valued algebro-geometric solutions of the two-component Camassa-Holm hierarchy
• Ghostpeakons and Characteristic Curves for the Camassa-Holm, Degasperis-Procesi and Novikov Equations