We establish the local well-posedness for a periodic two-component Camassa-Holm equation. We then present precise blow-up scenarios. Finally, we obtain several blow-up results and the blow-up rate of strong solutions to the equation.

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We first establish local well posedness for a weakly dissipative periodic 2-component Camassa-Holm system. We then present two global existence results for strong solutions to the equation. We finally obtain several blow-up results and the blow-up rate of strong solutions to the equation.

In this paper, we study the Cauchy problem and multi-soliton solutions for a two-component short pulse system. For the Cauchy problem, we first prove the existence and uniqueness of solution with an estimate of the analytic lifespan, and then investigate the continuity of the data-to-solution map in the space of analytic function. For the multi-soliton solutions, we first derive an N -fold Darboux transformation from the Lax pair of the two-component short pulse system, which is expressed in terms of the quasideterminant. Then by virtue of the N -fold Darboux transformation we obtain multi-loop and breather soliton solutions. In particular, one-, two-, three-loop soliton, and breather soliton solutions are discussed in details with interesting dynamical interactions and shown through figures.