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

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 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.