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

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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 Bˢₚ,ᵣ× Bˢₚ,ᵣ with 1≤ p,r ≤ +∞ and s>{2+1/p,5/2}. Moreover, we construct the local well-posedness in the critical Besov space B⁵/²₂,₁× B⁵/²₂,₁ and the blow-up criteria. In the paper, we also consider the well-posedness problem in the sense of Hadamard, non-uniform dependence, and Hölder 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 Hˢ× Hˢ,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 Hölder continuity in the Hʳ× Hʳ topology when 0≤ r< s with Hölder exponent α depending on both s and r.