Cauchy problem for a generalized cross-coupled Camassa-Holm system with waltzing peakons and higher-order nonlinearities

In this paper, we study the Cauchy problem for a generalized cross-coupled Camassa-Holm system with peakons and higher-order nonlinearities. By the transport equation theory and the classical Friedrichs regularization method, we obtain the local well-posedness of solutions for the system in nonhomogeneous Besov spaces $B^s_{p,r}\times B^s_{p,r}$ with $1\leq p,r \leq +\infty$ and $s>\max\{2+\frac{1}{p},\frac{5}{2}\}$. Moreover, we construct the local well-posedness in the critical Besov space $B^{5/2}_{2,1}\times B^{5/2}_{2,1}$ and the blow-up criteria. In the paper, we also consider the well-posedness problem in the sense of Hadamard, non-uniform dependence, and H\"older continuity of the data-to-solution map for the system on both the periodic and the non-periodic case. In light of a Galerkin-type approximation scheme, the system is shown well-posed in the Sobolev spaces $H^s\times H^s,s>5/2$ in the sense of Hadamard, that is, the data-to-solution map is continuous. However, the solution map is not uniformly continuous. Furthermore, we prove the H\"older continuity in the $H^r\times H^r$ topology when $0\leq r< s$ with H\"older exponent $\alpha$ depending on both $s$ and $r$.