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 ...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.展开更多
This article studies one dimensional viscous Camassa-Holm equation with a periodic boundary condition. The existence of the almost periodic solution is investigated by using the Galerkin method.
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, t...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.展开更多
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. Further...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.展开更多
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 ...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.展开更多
By constructing auxiliary differential equations, we obtain peaked solitary wave solutions of the generalized Camassa-Holm equation, including periodic cusp waves expressed in terms of elliptic functions.
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-...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.展开更多
文摘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.
基金Supported by Natural Science Foundation of China (10471047)Natural Science Foundation of Guangdong Province (05300162)
文摘This article studies one dimensional viscous Camassa-Holm equation with a periodic boundary condition. The existence of the almost periodic solution is investigated by using the Galerkin method.
基金Supported by the National Natural Science Foundation of China under Grant Nos 11571366 and 11501570the Open Foundation of State Key Laboratory of High Performance Computing of China+1 种基金the Research Fund of the National University of Defense Technology under Grant No JC15-02-02the Fund from HPCL
文摘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.
基金supported by NSFC(11471260)the Foundation of Shannxi Education Committee(12JK0850)
文摘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.
基金National Natural Science Foundation of China(Grants Nos.12271424 and 11871395)National Natural Science Foundation of China(Grants Nos.11971251,11631007 and 12111530003)。
文摘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.
基金Supported by the Nature Science Foundation of Shandong (No. 2004zx16,Q2005A01)
文摘By constructing auxiliary differential equations, we obtain peaked solitary wave solutions of the generalized Camassa-Holm equation, including periodic cusp waves expressed in terms of elliptic functions.
基金supported by National Natural Science Foundation of China(Grant Nos.11631007,11471174 and 11471259)。
文摘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.
