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.

A Multi-Symplectic Compact Method for the Two-Component Camassa-Holm Equation with Singular Solutions

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.

INVARIANT SUBSPACES AND GENERALIZED FUNCTIONAL SEPARABLE SOLUTIONS TO THE TWO-COMPONENT b-FAMILY SYSTEM

Invariant subspace method is exploited to obtain exact solutions of the two- component b-family system. It is shown that the two-component b-family system admits the generalized functional separable solutions. Furthermore, blow up and behavior of those exact solutions are also investigated.

Arbitrarily High-Order Energy-Preserving Schemes for the Camassa-Holm Equation Based on the Quadratic Auxiliary Variable Approach

In this paper,we present a quadratic auxiliary variable(QAV)technique to develop a novel class of arbitrarily high-order energy-preserving algorithms for the Camassa-Holm equation.The QAV approach is first utilized to transform the original equation into a reformulated QAV system with a consistent initial condition.Then the reformulated QAV system is discretized by applying the Fourier pseudo-spectral method in space and the symplectic Runge-Kutta methods in time,which arrives at a class of fully discrete schemes.Under the consistent initial condition,they can be rewritten as a new fully discrete system by eliminating the introduced auxiliary variable,which is rigorously proved to be energy-preserving and symmetric.Ample numerical experiments are conducted to confirm the expected order of accuracy,conservative property and efficiency of the proposed methods.The presented numerical strategy makes it possible to directly apply a special class of Runge-Kutta methods to develop energy-preserving algorithms for a general conservative system with any polynomial energy.

Orbital stability of two-component peakons

We prove that the two-component peakon solutions are orbitally stable in the energy space.The system concerned here is a two-component Novikov system,which is an integrable multi-component extension of the integrable Novikov equation.We improve the method for the scalar peakons to the two-component case with genuine nonlinear interactions by establishing optimal inequalities for the conserved quantities involving the coupled structures.Moreover,we also establish the orbital stability for the train-profiles of these two-component peakons by using the refined analysis based on monotonicity of the local energy and an induction method.

On the Cauchy problem and peakons of a two-component Novikov system

We study a two-component Novikov system,which is integrable and can be viewed as a twocomponent generalization of the Novikov equation with cubic nonlinearity.The primary goal of this paper is to understand how multi-component equations,nonlinear dispersive terms and other nonlinear terms affect the dispersive dynamics and the structure of the peaked solitons.We establish the local well-posedness of the Cauchy problem in Besov spaces B^s/p,r with 1 p,r+∞,s>max{1+1/p,3/2}and Sobolev spaces H^s(R)with s>3/2,and the method is based on the estimates for transport equations and new invariant properties of the system.Furthermore,the blow-up and wave-breaking phenomena of solutions to the Cauchy problem are studied.A blow-up criterion on solutions of the Cauchy problem is demonstrated.In addition,we show that this system admits single-peaked solitons and multi-peaked solitons on the whole line,and the single-peaked solitons on the circle,which are the weak solutions in both senses of the usual weak form and the weak Lax-pair form of the system.