Global Conservative Solutions of a Generalized Two-Component Camassa-Holm System

Periodic Conservative Solutions for the Two-component Camassa–holm System

We construct a global continuous semigroup of weak periodic conservative solutions to the two-component Camassa–Holm system, ut − utxx + κux + 3uux − 2uxuxx − uuxxx + ηρρx = 0 and ρt + (uρ)x = 0, for initial data (u, ρ)|t=0 in H1 per ×Lper. It is necessary to augment the system with an associated energy to identify the conservative solution. We study the stability of these periodic solutions by...

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

Global conservative solutions of the Camassa-Holm equation

This paper develops a new approach in the analysis of the Camassa-Holm equation. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose solutions are obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we ob...

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

In this paper we study a generalized two-component Camassa-Holm system which can be derived from the theory of shallow water waves moving over a linear shear flow. This new system also generalizes a class of dispersive waves in cylindrical compressible hyperelastic rods. We show that this new system can still exhibit the wave-breaking phenomenon. We also determine the exact blow-up rate of such...

Singular solutions of a modified two-component Camassa-Holm equation.

The Camassa-Holm (CH) equation is a well-known integrable equation describing the velocity dynamics of shallow water waves. This equation exhibits spontaneous emergence of singular solutions (peakons) from smooth initial conditions. The CH equation has been recently extended to a two-component integrable system (CH2), which includes both velocity and density variables in the dynamics. Although ...