Blow-up for a weakly dissipative modified two-component Camassa–Holm system

In this paper, we study the Cauchy problem of a weakly dissipative modified two-component Camassa–Holm (MCH2) system. We first derive the precise blow-up scenario and then give several criteria guaranteeing the blow-up of the solutions. We finally discuss the blow-up rate of the blowing-up solutions.     

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

Considered herein is the well-posedness problem of the periodic two-component Dullin–Gottwald–Holm (DGH) system on the circle, which can be derived from Euler?s equation with constant vorticity in shallow water waves moving over a linear shear flow. The result of blow-up solutions for certain initial profiles in a manner which corresponds to wave-breaking is established.     

Wave breaking for a modified two-component Camassa–Holm system

In this paper, we establish sufficient conditions on the initial data to guarantee blow-up phenomenon for the modified two-component Camassa–Holm (MCH2) system.     

Blow-up criteria of solutions to a modified two-component Camassa–Holm system

In this paper, we consider a modified two-component Camassa–Holm (MCH2) system which arises in shallow water theory. We analyze the wave breaking mechanism by establishing some new blow-up criteria for this system formulated either on the line or with space-periodic initial condition.     

Stability of singular waves for Dullin–Gottwald–Holm equation

In this paper, the asymptotic stability of the solutions near the explicit singular waves of Dullin–Gottwald–Holm equation is studied based on the commutator estimate, the semi-group theory of linear operator and the Banach fixed point theorem.     

Global weak solutions for the Dullin–Gottwald–Holm equation

We prove the existence and uniqueness of global weak solutions to the Dullin–Gottwald–Holm equation provided the initial data satisfies certain conditions.     

Global existence and blow-up phenomena for a weakly dissipative 2-component Camassa–Holm system

We study the Cauchy problem of a weakly dissipative 2-component Camassa–Holm system. We first establish local well-posedness for a weakly dissipative 2-component Camassa–Holm system. We then present a global existence result for strong solutions to the system. We finally obtain several blow-up results and the blow-up rate of strong solutions to the system.     

On the solutions of the Dullin–Gottwald–Holm equation in Besov spaces

This paper is concerned with the Cauchy problem for the Dullin–Gottwald–Holm equation. First, the local well-posedness for this system in Besov spaces is established. Second, the blow-up criterion for solutions to the equation is derived. Then, the existence and uniqueness of global solutions to the equation are investigated. Finally, the sharp estimate from below and lower semicontinuity for the existence time of solutions to this equation are presented.     

Orbital stability of the sum of N peakons for the Dullin–Gottwald–Holm equation

Analogous to the Camassa–Holm equation, the Dullin–Gottwald–Holm equation also possesses peaked solitary waves, which are called peakons. We prove in this paper the stability of ordered trains of peakons for the Dullin–Gottwald–Holm equation.     

Well-posedness and blow-up solution for a modified two-component periodic Camassa–Holm system with peakons

Considered herein is a modified two-component periodic Camassa–Holm system with peakons. The local well-posedness and low regularity result of solutions are established. The precise blow-up scenarios of strong solutions and several results of blow-up solutions with certain initial profiles are described in detail and the exact blow-up rate is also obtained.     

Well-posedness and blow-up for a modified two-component Camassa–Holm equation

In this article, we consider a newly modified two-component Camassa–Holm equation. First, we establish the local well-posedness result, then we present a precise blow-up scenario. Afterwards, we derive a new conservation law, by which and the precise blow-up scenario we prove three blow-up results and a blow-up rate estimate result.     

A remark on wave breaking for the Dullin–Gottwald–Holm equation

In this paper, a new sufficient condition to guarantee wave breaking for the Dullin–Gottwald–Holm equation is established, which is a local criterion and easy to check.     

Variational derivation of two-component Camassa–Holm shallow water system

By a variational approach in the Lagrangian formalism, we derive the nonlinear integrable two-component Camassa–Holm system (1). We show that the two-component Camassa–Holm system (1) with the plus sign arises as an approximation to the Euler equations of hydrodynamics for propagation of irrotational shallow water waves over a flat bed. The Lagrangian used in the variational derivation is not a metric.     

On Solitary Waves and Wave-Breaking Phenomena for a Generalized Two-Component Integrable Dullin–Gottwald–Holm System

We study here a generalized two-component integrable Dullin–Gottwald–Holm system, which can be derived from the Euler equation with constant vorticity in shallow water waves moving over a linear shear flow. We first derive this system in the shallow-water regime. We next classify all traveling wave solution of this system. Finally, we study the blow-up mechanism and give two sufficient conditions which can guarantee wave-breaking phenomena.