Stability of Solitary Waves and Wave-Breaking Phenomena for the Two-Component Camassa-Holm System

Abstract

Considered herein is a two-component Camassa-Holm system modeling shallow water waves moving over a linear shear flow. It is shown here that solitary-wave solutions of the system are dynamically stable to perturbations for a range of their speeds. On the other hand, a new wave-breaking criterion for solutions is established and two results of wave-breaking solutions with certain initial profiles are described in detail. Moreover, a sufficient condition for global solutions determined only by a nonzero initial profile of the free surface component of the system is found.

On the wave-breaking phenomena for the two-component Dullin-Gottwald-Holm system

“…Obviously, if ρ ≡ 0, then (1.6) becomes the Camassa-Holm equation (1.3). Many recent works are devoted to studying system (1.6), for instance; see [11,18,[20][21][22][23][24][25]30] and the references therein.…”

Wave Breaking and Measure of Momentum Support for an Integrable Camassa‐Holm System with Two Components

“…If the initial value (ρ 0 (x), y 0 (x)) has compact support, then (y(·, t), ρ(·, t)) is also possible to be compactly supported due to (12) and (13). Moreover, we find that the supports of y and ρ propagate along the flow.…”

“…The extended supersymmetric Camassa‐Holm equation was also presented by Z. Popowicz in . Stability of traveling wave solutions and wave breaking phenomenon are the subjects of . Very recently, smooth traveling wave solutions with was investigated by Mustafa in, the persistence properties of strong solutions were discussed in and the investigation of other related two‐component models can be found in .…”

A Lagrangian View on Complete Integrability of the Two-Component Camassa–Holm System

“…This means that the spatial derivative u x ( · , t) of the solution (u( · , t), ρ( · , t)) becomes unbounded within finite time, while both u( · , t) H 1 (R) and ρ( · , t) L 2 (R) remain bounded, see e.g. [27,28,29,30,43,44] and the references therein. In addition, energy concentrates on sets of measure zero.…”