Wave-Breaking and Global Existence for a Generalized Two-Component Camassa-Holm System

Stability Of Solitary Waves Of A Generalized Two-Component Camassa-Holm System

We study here the existence of solitary wave solutions of a generalized twocomponent Camassa-Holm system. In addition to those smooth solitary-wave solutions, we show that there are solitary waves with singularities: peaked and cusped solitary waves. We also demonstrate that all smooth solitary waves are orbitally stable in the energy space. We finally give a sufficient condition for global str...

Stability of Solitary Waves and Global Existence of a Generalized Two-Component Camassa–Holm System

Stability of the Camassa-Holm peakons in the dynamics of a shallow-water-type system

The stability of the Camassa-Holm (periodic) peakons in the dynamics of an integrable shallow-water-type system is investigated. A variational approach with the use of the Lyapunov method is presented to prove the variational characterization and the orbital stability of these wave patterns. In addition, a sufficient condition for the global existence of strong solutions is given. Finally, a lo...

Global Dissipative Solutions of the Camassa-Holm Equation

This paper is concerned with the global existence of dissipative solutions to the Camassa-Holm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as an O.D.E. in an L∞ space, containing a non-local source term which is discontinuous but has bounded directional variation along a suitable cone of directions. For a give...

Global Weak Solutions to a Generalized Hyperelastic-rod Wave Equation

We consider a generalized hyperelastic-rod wave equation (or generalized Camassa– Holm equation) describing nonlinear dispersive waves in compressible hyperelastic rods. We establish existence of a strongly continuous semigroup of global weak solutions for any initial data from H1(R). We also present a “weak equals strong”uniqueness result.