Two-component Camassa Holm Shallow Water System Research Articles

Global existence and wave breaking for a stochastic two-component Camassa–Holm system

In this paper, we study the stochastic two-component Camassa–Holm shallow water system on R and T≔R/2πZ. We first establish the existence, uniqueness, and blow-up criterion of the pathwise strong solution to the initial value problem with nonlinear noise. Then, we consider the impact of noise on preventing blow-up. In both nonlinear and linear noise cases, we establish global existence. In the nonlinear noise case, the global existence holds true with probability 1 if a Lyapunov-type condition is satisfied. In the linear noise case, we provide a lower bound for the probability that the solution exists globally. Furthermore, in the linear noise and the periodic case, we formulate a precise condition on initial data that leads to blow-up of strong solutions with a positive probability, and the lower bound for this probability is also estimated.

Stability of the train of N solitary waves for the two-component Camassa–Holm shallow water system

Considered herein is the integrable two-component Camassa–Holm shallow water system derived in the context of shallow water theory, which admits blow-up solutions and the solitary waves interacting like solitons. Using modulation theory, and combining the almost monotonicity of a local version of energy with the argument on the stability of a single solitary wave, we prove that the train of N solitary waves, which are sufficiently decoupled, is orbitally stable in the energy space H1(R)×L2(R).

Persistence properties of the solutions to a generalized two-component Camassa–Holm shallow water system

This paper is concerned with the Cauchy problem for a generalized two-component Camassa–Holm shallow water system. We prove that the solution will maintain the corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively.

On the Global Dissipative and Multipeakon Dissipative Behavior of the Two-Component Camassa-Holm System

The global dissipative and multipeakon dissipative behavior of the two-component Camassa-Holm shallow water system after wave breaking was studied in this paper. The underlying approach is based on a skillfully defined characteristic and a set of newly introduced variables which transform the original system into a Lagrangian semilinear system. It is the transformation, together with the associated properties, that allows for the continuity of the solution beyond collision time to be established, leading to a uniquely global dissipative solution, which constructs a semigroup, and the multipeakon dissipative solution.

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.

Global conservative solutions of a modified two-component Camassa–Holm shallow water system

We first prove the existence of global conservative solutions to the Cauchy problem for a modified two-component Camassa–Holm shallow water system. Then, we show that these global solutions, which depend continuously on the initial date, construct a semigroup.

ON THE LOW REGULARITY SOLUTIONS FOR A MODIFIED TWO-COMPONENT CAMASSA–HOLM SHALLOW WATER SYSTEM

AbstractWe first study the regularised version of a modified two-component Camassa–Holm shallow water system and obtain the energy estimates of the corresponding approximate solutions. Then, we present a sufficient condition which guarantees that these approximate solutions converge to a low regularity weak solution of the modified two-component Camassa–Holm shallow water system.

Global dissipative solutions of a modified two-component Camassa–Holm shallow water system

By introducing a new set of independent variables, we first transform a modified two-component Camassa–Holm shallow water system into a semilinear system. To obtain a dissipative solution, we modify the corresponding system into a discontinuous system. Then we map the solution of system to the dissipative solution of original equation. Moreover we prove that these global dissipative solutions construct a semigroup.

Global weak solutions for a two-component Camassa–Holm shallow water system

In this paper, we prove the existence of global weak solution for an integrable two-component Camassa–Holm shallow water system provided the initial data satisfying some certain conditions.

Analytic solutions of the Cauchy problem for two-component shallow water systems

This paper is mainly concerned with the Cauchy problem for a modified two-component Camassa–Holm shallow water system with analytic initial data. The analyticity of its solutions is proved in both variables, globally in space and locally in time. The obtained results can be also applied to the classical integrable two-component Camassa–Holm shallow water system and the b-equation.

The Generalized Hunter–Saxton System

A new system of equations generalizing the Hunter–Saxton system [Wunsch, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), pp. 647–656; Constantin and Ivanov, On an integrable two-component Camassa–Holm shallow water system, Phys. Lett. A, 372 (2008), pp. 7129–7132] is derived and investigated. In the periodic setting, we show that there exist local solutions for which we provide sufficient conditions in order that they stay bounded or blow up in finite time; on the real line, we construct a class of global weak solutions.

Global existence and blow-up phenomena for an integrable two-component Camassa–Holm shallow water system

This paper is concerned with global existence and blow-up phenomena for an integrable two-component Camassa–Holm shallow water system. A new global existence result and several new blow-up results of strong solutions to the system are presented. Our obtained results for the system are sharp and improve considerably earlier results.

On smooth traveling waves of an integrable two-component Camassa–Holm shallow water system

We give a simple proof of existence for the smooth traveling waves with a single crest profile of maximum amplitude of a recently derived integrable two-component shallow water system.

On an integrable two-component Camassa–Holm shallow water system

The interest in the Camassa–Holm equation inspired the search for various generalizations of this equation with interesting properties and applications. In this Letter we deal with such a two-component integrable system of coupled equations. First we derive the system in the context of shallow water theory. Then we show that while small initial data develop into global solutions, for some initial data wave breaking occurs. We also discuss the solitary wave solutions. Finally, we present an explicit construction for the peakon solutions in the short wave limit of system.