Two-component Camassa Holm Equation Research Articles

Dependence on initial data for a stochastic modified two-component Camassa-Holm system

We study a stochastic modified two-component Camassa-Holm equation on R. We establish a local well-posedness result in the sense of Hadamard, i.e. existence, uniqueness and continuous dependence on initial data, as well as blow-up criteria for pathwise solutions in the Sobolev spaces Hs with s>3/2. Motivated by the work of Miao et al. (2024) [29], we show that the solution map y0↦y(t) defined by the corresponding Cauchy problem is weakly unstable, due to either a lack of strong stability in the exiting time or the absence of uniformly continuous dependence on the initial data.

The multi-soliton solutions of another two-component Camassa–Holm equation with Darboux transformation approach

In this paper, we develop another approach to construct the multi-soliton solutions of a two-component Camassa–Holm equation in terms of Wronskians with help of a reciprocal transformation and a gauge transformation. Its kink solution, loop solution and smooth soliton solution are presented. Then with the non-trivial limiting procedure, the solution of Camassa–Holm equation is also derived from that of two-component Camassa–Holm equation.

Wave-breaking and weak instability for the stochastic modified two-component Camassa–Holm equations

Wave-breaking and weak instability for the stochastic modified two-component Camassa–Holm equations

Multisoliton solutions of the two-component Camassa–Holm equation and its reductions

Multisoliton solutions of the two-component Camassa–Holm equation and its reductions

Multisoliton solutions of the two-component Camassa-Holm equation and its reductions

Анализируются преобразования Беклунда для интегрируемого двухкомпонентного уравнения Камассы-Холма (2КХ), в которые входят как зависимые, так и независимые переменные. Получена формула нелинейной суперпозиции для построения многосолитонного, многопетлевого и многокинкового решений уравнения 2КХ. Предъявлены решения уравнения Камассы-Холма, двухкомпонентного уравнения Хантера-Сакстона (2ХС) и уравнения Хантера-Сакстона, все они возникают из решений уравнения 2КХ. В частности, с помощью перехода к соответствующему пределу решение уравнения 2ХС можно успешно получить из решения уравнения 2КХ, это показано методом преобразования Беклунда. Путем анализа решения получены солитонное и петлевое решения уравнения 2ХС.

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.

On some integrability aspects of the two-component Camassa–Holm equation

The integrability of the two-component Camassa–Holm shallow water equation with the gravity acceleration points downwards and upwards is investigated by prolongation structure theory and the geometric approach of pseudo-spherical surfaces. It is shown that this equation is completely integrable in sense of having Lax pair and describing the pseudo-spherical surface. The reciprocal transformation is proposed to connect its solution with the solution of a conservative equation.

Geometric Flows of Curves, Two-Component Camassa-Holm Equation and Generalized Heisenberg Ferromagnet Equation

In this paper, we study the generalized Heisenberg ferromagnet equation, namely, the M-CVI equation. This equation is integrable. The integrable motion of the space curves induced by the M-CVI equation is presented. Using this result, the Lakshmanan (geometrical) equivalence between the M-CVI equation and the two-component Camassa-Holm equation is established.

Bifurcations and Exact Traveling Wave Solutions of Two Shallow Water Two-Component Systems

This paper studies two two-component shallow water wave models. From the dynamical systems approach and using the singular traveling wave theory developed by Li and Chen [2007], all possible bounded solutions (solitary wave solutions, pseudo-peakons, periodic peakons, as well as smooth periodic wave solutions) are obtained under different parameter conditions. More than six explicit exact parametric representations are derived. More interestingly, it was found that, for the two-component Camassa–Holm equations with constant vorticity, its [Formula: see text]-traveling wave system has a pseudo-peakon wave solution. In addition, its [Formula: see text]-traveling wave system has four families of uncountably infinitely many solitary wave solutions. The new results complete a recent study of Dutykh and Ionescu-Kruse [2019].

Blow-up for two-component Camassa-Holm equation with generalized weak dissipation

<p style='text-indent:20px;'>This paper is concerned with blow-up solution for the Cauchy problem of two-component Camassa-Holm equation with generalized weak dissipation. By Kato's theorem and monotonicity, we investigate the local well-posedness of Cauchy problem and establish the blow-up criteria and the blow-up rate. Moreover, the property of blow-up points set is characterized.

On equivalence of one spin system and two-component Camassa-Holm equation

On equivalence of one spin system and two-component Camassa-Holm equation

Blow-Up for a Periodic Two-Component Camassa-Holm Equation with Generalized Weakly Dissipation

In this paper we study a periodic two-component Camassa-Holm equation with generalized weakly dissipation. The local well-posedness of Cauchy problem is investigated by utilizing Kato’s theorem. The blow-up criteria and the blow-up rate are established by applying monotonicity. Finally, the global existence results for solutions to the Cauchy problem of equation are proved by structuring functions.

Bifurcations and Exact Solutions of Generalized Two-Component Peakon Type Dual Systems

This paper investigates two generalized two-component peakon type dual systems, which can be reduced to the same planar dynamical systems via the dynamical system approach and the theory of singular traveling wave systems, where one of them contains the two-component Camassa–Holm system. By bifurcation analysis on the corresponding traveling wave system, we obtain the phase portraits and derive possible exact traveling wave solutions that include solitary wave solution, peakon and anti-peakon, pseudo-peakon, periodic peakon, compacton and periodic wave solution. Our results are also applicable to the two-component Camassa–Holm equation.

Global well-posedness for the two-component Camassa–Holm equation with fractional dissipation

In this paper, we investigate the two-component Camassa–Holm equation with fractional dissipation [Formula: see text] where [Formula: see text] denotes the fractional dissipative operator which is defined by the Fourier transform [Formula: see text]. We prove the local-posedness of this equation via the Littlewood–Paley theory and the suitable iterative scheme. Furthermore, under appropriate discussions, we give the global well-posedness of the above equation.

Effects of vorticity on the travelling waves of some shallow water two-component systems

In the present study we consider three two-component (integrable and non-integrable) systems which describe the propagation of shallow water waves on a constant shear current. Namely, we consider the two-component Camassa-Holm equations, the Zakharov-Ito system and the Kaup--Boussinesq equations all including constant vorticity effects. We analyze both solitary and periodic-type travelling waves using the simple and geometrically intuitive phase space analysis. We get the pulse-type solitary wave solutions and the front solitary wave solutions. For the Zakharov-Ito system we underline the occurrence of the pulse and anti-pulse solutions. The front wave solutions decay algebraically in the far field. For the Kaup-Boussinesq system, interesting analytical multi-pulsed travelling wave solutions are found.

Numerical solutions to a two-component Camassa–Holm equation

Abstract In the present paper, a three-step iterative algorithm for solving a two-component Camassa–Holm (2CH) equation is presented. In the first step, the time-dependent equation for the horizontal fluid velocity with nonlinear convection is solved. Then an inhomogeneous Helmholtz equation is solved. Finally, the equation for modeling the transport of density is solved in the third step. The differential order of 2CH equation has been reduced in order to facilitate numerical scheme development in a comparatively smaller grid stencil. In this study, a fifth-order spatially accurate upwinding combined compact difference scheme (UCCD5) which differs from that in Sheu et al. (2011) is developed in a four-point grid stencil for approximating the first-order derivative term. For the purpose of retaining long-time Hamiltonians in the 2CH equation, the time integrator (or time-stepping scheme) chosen is symplectic. Various numerical experiments such as the single peakon, peakon–antipeakon interaction and dam-break problems are conducted to illustrate the effectiveness of the proposed numerical method. It is shown that both the Hamiltonians and Casimir functions are conserved well for all problems.

Liouville theorems for periodic two-component shallow water systems

We establish Liouville-type theorems for periodic two-component shallow water systems, including a two-component Camassa-Holm equation (2CH) and a two-component Degasperis-Procesi (2DP) equation. More presicely, we prove that the only global, strong, spatially periodic solutions to the equations, vanishing at some point $(t_0, x_0)$, are the identically zero solutions. Also, we derive new local-in-space blow-up criteria for the dispersive 2CH and 2DP.

A Multi-Symplectic Compact Method for the Two-Component Camassa–Holm Equation with Singular Solutions**Supported by the National Natural Science Foundation of China under Grant Nos 11571366 and 11501570, the Open Foundation of State Key Laboratory of High Performance Computing of China, the Research Fund of the National University of Defense Technology under Grant No JC15-02-02, and the Fund from HPCL.

The two-component Camassa–Holm equation includes many intriguing phenomena. We propose a multi-symplectic compact method to solve the two-component Camassa–Holm equation. Based on its multi-symplectic formulation, the proposed method is derived by the sixth-order compact finite difference method in spatial discretization and the symplectic implicit midpoint scheme in temporal discretization. Numerical experiments finely describe the velocity and density variables in the two-component integrable system and distinctly display the evolvement of the singular solutions. Moreover, the proposed method shows good conservative properties during long-time numerical simulation.

Well-posedness and persistence properties for two-component higher order Camassa–Holm systems with fractional inertia operator

In this paper, we study the Cauchy problem for a two-component higher order Camassa–Holm systems with fractional inertia operator A=(1−∂x2)r,r≥1, which was proposed by Escher and Lyons (2015). By the transport equation theory and Littlewood–Paley decomposition, we confirm the local well-posedness of solutions for the system in nonhomogeneous Besov spaces Bp,qs×Bp,qs−2r+1 with 1≤p,q≤+∞ and the Besov index s>max{2r+1p,2r+1−1p}. Moreover, we demonstrate the local well-posedness in the critical Besov space B2,12r+12×B2,132. On the other hand, the propagation behavior of compactly supported solutions is examined, namely whether solutions which are initially compactly supported will retain this property throughout their time of evolution. Finally, we also establish the persistence properties of the solutions to the two-component Camassa–Holm equation with r=1 in weighted Lϕp:=Lp(R,ϕp(x)dx) spaces for a large class of moderate weights.

Multi-soliton solution to the two-component Hunter–Saxton equation

In this paper, we study the bilinear form and the general N-soliton solution for a two-component Hunter–Saxton (2-HS) equation, which is the short wave limit of a two-component Camassa–Holm equation. By defining a hodograph transformation based on a conservation law and appropriate dependent variable transformations, we propose a set of bilinear equations which yields the 2-HS equation. Furthermore, we construct the N-soliton solution to the 2-HS equation based on the tau functions of an extended two-dimensional Toda-lattice hierarchy through reductions. One- and two-soliton solutions are calculated and analyzed.