Peakon weak solutions for the rotation-two-component Camassa–Holm system

Recently, Fan, Gao and Liu proposed a kind of rotation-two-component Camassa–Holm system. In this paper, we investigate whether the rotation-two-component Camassa–Holm system admits peakon-delta weak solutions in distribution sense. As special reductions, all peakon solutions for generalized Dullin–Gottwald–Holm system, two-component Camassa–Holm system, Dullin–Gottwald–Holm equation and Camassa–Holm equation are recovered from the corresponding results of rotation-two-component Camassa–Holm system.     

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.     

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.     

Global weak solutions for a periodic two-component?μ-Hunter–Saxton system

This paper is concerned with global existence of weak solutions for a periodic two-component?μ-Hunter–Saxton system. We first derive global existence for strong solutions to the system with smooth approximate initial data. Then, we show that the limit of approximate solutions is a global weak solution of the two-component?μ-Hunter–Saxton system.     

Algebro-geometric solutions for the two-component Camassa–Holm Dym hierarchy

This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Camassa–Holm Dym (CHD2) hierarchy. Our main tools include the polynomial recursive formalism, the hyperelliptic curve with finite number of genus, the Baker–Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the algebro-geometric solutions are obtained for the entire CHD2 hierarchy.     

Global solutions for the generalized Camassa–Holm equation

In this paper, we study the Cauchy problem of the generalized Camassa–Holm equation. Firstly, we prove the existence of the global strong solutions provide the initial data satisfying a certain sign condition. Then, we obtain the existence and the uniqueness of the global weak solutions under the same sign condition of the initial data.     

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.     

Global attractor for the viscous modified two-component Camassa–Holm equation

We obtain the existence of global attractor for the Cauchy problem of a viscous modified two-component Camassa–Holm equation. The existence of global strong solutions is obtained using Kato’s theory. The key elements in our analysis are the uniform Gronwall lemma and some estimates of the solutions.     

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.     

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.     

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.     

Dissipative solutions for the modified two–component Camassa–Holm system

Camassa–Holm model is capable of characterizing the dynamic behavior of shallow water wave, thus has been extensively studied. This paper is concerned with shallow water wave behavior after wave breaking. To better reflect the whole process, the modified two-component Camassa–Holm system is considered. The continuation of solutions of such system after wave braking is investigated. By introducing a skillfully defined characteristic, together with a set of newly defined variables, the original system is converted into a Lagrangian equivalent system, from which global dissipative solutions are obtained. The results obtained herein are deemed useful in understanding the dynamic behavior of shallow water wave during and after wave breaking.     

Initial boundary value problem for a coupled Camassa–Holm system with peakons

We investigate the homogeneous initial boundary value problem for a coupled Camassa–Holm system with peakons on the half line. We first establish the local well-posedness for the system. We then present a precise blowup scenario and several blowup results of strong solutions to the system. We finally give the blowup rate of strong solutions to the system when blowup occurs.     

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 cross-coupled Camassa–Holm system

In this paper, we study the Cauchy problem for a recently derived system of two cross-coupled Camassa–Holm equations. We firstly establish the local well-posedness result of this system in Besov spaces by using Littlewood–Paley decomposition and the transport equation theory, and then present a precise blow-up scenario for strong solutions.     

Global Solutions for the Two-Component Camassa–Holm System

We prove existence of a global conservative solution of the Cauchy problem for the two-component Camassa–Holm (2CH) system on the line, allowing for nonvanishing and distinct asymptotics at plus and minus infinity. The solution is proven to be smooth as long as the density is bounded away from zero. Furthermore, we show that by taking the limit of vanishing density in the 2CH system, we obtain the global conservative solution of the (scalar) Camassa–Holm equation, which provides a novel way to define and obtain these solutions. Finally, it is shown that while solutions of the 2CH system have infinite speed of propagation, singularities travel with finite speed.