Two-component Camassa Holm System 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.

Multi-Symplectic Method for the Two-Component Camassa–Holm (2CH) System

In this paper, the multi-symplectic formulations of the two-component Camassa–Holm system are presented. Both the multi-symplectic structure and two local conservation laws of the generalized two-component Camassa–Holm model are proposed for its first-order canonical form. Then, combining the Fourier pseudo-spectral method in the spatial domain with the midpoint method in the time dimension, the multi-symplectic Fourier pseudo-spectral scheme is constructed for the first-order canonical form. Meanwhile, the discrete scheme of the residuals of the multi-symplectic structure and two local conservation laws are also provided. By using the multi-symplectic Fourier pseudo-spectral scheme, the evolution of one- and two-soliton solutions for the generalized two-component Camassa–Holm model is regained. The structure-preserving properties and the reliability of the numerical scheme are illustrated by the tiny numerical residuals (less than 3.5 × 10−8) of the conservation laws as well as the tiny numerical variations (less than 1 × 10−9) of the amplitudes and the propagating velocities of the solitons.

A Generalized Two-Component Camassa-Holm System with Complex Nonlinear Terms and Waltzing Peakons

In this paper, we deal with the Cauchy problem for a generalized two-component Camassa-Holm system with waltzing peakons and complex higher-order nonlinear terms. By the classical Friedrichs regularization method and the transport equation theory, the local well-posedness of solutions for the generalized coupled Camassa-Holm system in nonhomogeneous Besov spaces and the critical Besov space B^{5/2}_{2,1}times B^{5/2}_{2,1} was obtained. Besides the propagation behaviors of compactly supported solutions, the global existence and precise blow-up mechanism for the strong solutions of this system are determined. In addition to wave breaking, the another one of the most essential property of this equation is the existence of waltzing peakons and multi-peaked solitray was also obtained.

The well-posedness for the Camassa-Holm type equations in critical Besov spaces [formula omitted] with 1 ≤ p < +∞

For the famous Camassa-Holm equation, the well-posedness in C([0,T];Bp,11+1p(R)) with 1≤p≤2 and the ill-posedness in C([0,T];B∞,11(R)) had been studied in [15,16,23]. That is to say, it left an open problem in the critical case C([0,T];Bp,11+1p(R)) with 2<p<+∞ proposed by Danchin in [15,16]. In this paper, we solve this problem and obtain the local well-posedness for the Camassa-Holm equation in critical Besov spaces C([0,T];Bp,11+1p(R)) with 1≤p<+∞. It is worth mentioning that our method is suitable for many Camassa-Holm type equations, such as the Novikov equation and the two-component Camassa-Holm system, and can also improve their index of the local well-posedness.

Blow-up data for a two-component Camassa-Holm system with high order nonlinearity

This paper is concerned with the Cauchy problem for a two-component Camassa-Holm system with high order nonlinearity, which is a multi-component extension of the Fokas-Olver-Rosenau-Qiao equation. Firstly, we state the local well-posedness for the Cauchy problem of the system in the framework of Sobolev-Besov spaces. Then, we establish the precise blow-up mechanism for the strong solutions by means of the transport equation theory. As is well-known, the H1-norm conservation law of the velocity component is crucial to study blow-up phenomena of the single Camassa-Holm type equation. However, it is no longer available for our nonlinear coupled system. To overcome this difficulty, we derive some suitable conservation laws by sufficiently exploiting the fine structure of the system. Based on which, we finally construct several new blow-up strong solutions with certain initial profiles in finite time.

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.

Non-uniform continuity on initial data for the two-component b-family system in Besov space

This paper studies a two-component b-family system, which includes the two-component Camassa-Holm system and the two-component Degasperis-Procesi system as special case. It is shown that the solution map of this system is not uniformly continuous on the initial data in Besov spaces \(B_{p, r}^{s-1}({\mathbb {R}})\times B_{p, r}^s({\mathbb {R}})\) with \(s>\max \{1+\frac{1}{p}, \frac{3}{2}\}\), \(1\le p, r< \infty \). Our result covers and extends the previous non-uniform continuity in Sobolev spaces \(H^{s-1}({\mathbb {R}})\times H^s({\mathbb {R}})\) for \(s>\frac{5}{2}\) to Besov spaces (Nonlinear Anal., 2014, 111: 1-14). Compared with the generalized rotation b-family system considered by Holmes et al. (Z. Angew. Math. Mech., 2021), our non-uniform continuity is established in a broader range of Besov spaces.

On the stochastic two-component Camassa-Holm system driven by pure jump noise

We consider the stochastic two-component Camassa-Holm system driven by pure jump noise in the Marcus canonical form. Using the regularization method, the stochastic compactness method and the stochastic renormalized formulations, we prove the existence of global martingale solutions for the stochastic two-component Camassa-Holm system.

On the absolutely continuous spectrum of generalized indefinite strings II

We continue to investigate absolutely continuous spectrum of generalized indefinite strings. By following an approach of Deift and Killip, we establish stability of the absolutely continuous spectra of two more model examples of generalized indefinite strings under rather wide perturbations. In particular, one of these results allows us to prove that the absolutely continuous spectrum of the isospectral problem associated with the two-component Camassa-Holm system in a certain dispersive regime is essentially supported on the set $(-\infty,-1/2]\cup [1/2,\infty)$.

Global Existence and Blow-Up for a Weakly Dissipative Modified Two-Component Camassa-Holm System

In this paper, we consider a weakly dissipative modified two-component Camassa-Holm system. We demonstrate a simple sufficient condition on initial data to guarantee blow-up of solutions in finite time, and the solutions exist globally in time.

On Blow-Up of Solutions to a Weakly Dissipative Two-Component Camassa–Holm System

In this paper, we consider a weakly dissipative two-component Camassa–Holm system. We demonstrate a simple sufficient condition on initial date to guarantee blow-up of solutions in finite time and guarantee the solutions exist globally in time. The results improve considerable the previous results.

Blow-up criteria for a two-component nonlinear dispersive wave system

We investigate the blow-up phenomena for the two-component generalizations of nonlinear dispersive wave equation on the real line that includes the two-component Camassa-Holm system as well as the system for the hyperelastic-rod wave equation as special cases. We establish a blow-up criterion for system of coupled equations under certain natural initial profiles. The obtained criterion involves only the properties of the data u0 in a neighbourhood of a single point and consequently, it is local-in-space with respect to first component of the system under consideration.

Generalized Nonlocal Symmetries of Two-Component Camassa–Holm and Hunter–Saxton Systems

The two-component Camassa–Holm system and two-component Hunter–Saxton system are completely integrable models. In this paper, it is shown that these systems admit nonlocal symmetries by their geometric integrability. As an application, we obtain the recursion operator and conservation laws by using this kind of nonlocal symmetries.

Blow-up for an integrable two-component Camassa-Holm system with cubic nonlinearity and peakon solutions

Blow-up for an integrable two-component Camassa-Holm system with cubic nonlinearity and peakon solutions

Local-in-Space Blow-up for a Weakly Dissipative Generalized Two-Component Camassa–Holm System

Local-in-Space Blow-up for a Weakly Dissipative Generalized Two-Component Camassa–Holm System

On wave-breaking phenomena for a new generalized two-component shallow water wave system

In this paper, we mainly devote to investigate a generalized two-component Camassa–Holm system which can be derived from shallow water wave in equatorial ocean regions by Hu and Liu (Phys D 391:87–110, 2019). Depending on different real-valued interval of the competition or balance index $$\sigma $$ , we establish the new wave-breaking criteria and extend the earlier blow-up results for the system.

Global-in-time solvability and blow-up for a non-isospectral two-component cubic Camassa-Holm system in a critical Besov space

In this paper, we prove the global Hadamard well-posedness of strong solutions to a non-isospectral two-component cubic Camassa-Holm system in the critical Besov space B2,11/2(T). Our results show that in comparison with the well-known work for classic Camassa-Holm-type equations, the existence of global solution only relies on the L1-integrability of the variable coefficients α(t) and γ(t), but nothing to do with the shape of the initial data. The key ingredient of the proof hinges on the careful analysis of the mutual effect among two component forms, the uniform bound of approximate solutions, and several crucial estimates of cubic nonlinearities in low-regularity Besov spaces via the Littlewood-Paley decomposition theory. A reduced case in our results yields the global existence of solutions in a Besov space for two kinds of well-known isospectral peakon system with weakly dissipative terms. Moreover, we derive two kinds of precise blow-up criteria for a strong solution in both critical and non-critical Besov spaces, as well as providing specific characterization for the lower bound of the blow-up time, which implies the global existence with additional conditions on the time-dependent parameters α(t) an γ(t). We believe that the method in this paper can be applied to the study of well-posedness for other transport type equations.

Multi-soliton solutions of a two-component Camassa–Holm system: Darboux transformation approach

We propose and develop another approach to constructing multi-soliton solutions of an integrable two-component Camassa–Holm (CH2) system. With the help of a reciprocal transformation and a gauge transformation, we relate the CH2 system to a negative flow of the Broer–Kaup or two-boson hierarchy. The solutions of this negative flow are given in terms of Wronskians via Darboux transformation. Then the multi-soliton solutions of the CH2 system are recovered in parametric form by inverting the reciprocal transformation and the gauge transformation.

The global Gevrey regularity of the rotation two-component Camassa-Holm system

In this paper, we study the rotation two-component Camassa-Holm system, which is a model in the equatorial water waves with the effect of the Coriolis force. We establish the global Gevrey regularity of the rotation two-component Camassa-Holm system in Gevrey class Gτ with τ≥1 in time. Our obtained result improves considerably the recent result in [45].

On the blow up solutions to a two-component cubic Camassa-Holm system with peakons

This paper is concerned with the Cauchy problem for a two- component cubic Camassa-Holm system with peakons, which is a natural multi-component extension of the single Fokas-Olver-Rosenau-Qiao equation. By sufficiently exploiting the fine structure of the system, we derive two useful conservation laws which turns out an exponential increase estimate for the \begin{document}$ L^\infty $\end{document} -norm of the strong solution within its lifespan. As a result, two new blow-up solutions with certain initial profiles are established.