Continuity for a generalized cross-coupled Camassa–Holm system with waltzing peakons and higher-order nonlinearities. (February 2020) - Zhou, Shouming; Qiao, Zhijun; Mu, Chunlai
Abstract: In this paper, we consider the well-posedness problem in the sense of Hadamard, non-uniform dependence, and Hölder continuity of the data-to-solution map for a generalized cross-coupled Camassa–Holm system with waltzing peakons 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 × 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 Hölder continuity in the H r × H r topology when 0 ≤ r < s with Hölder exponent α depending on both s and r .