The periodic Cauchy problem for a combined CH–mCH integrable equation

This paper is concerned with the periodic Cauchy problem for a generalized Camassa–Holm integrable equation, which can be viewed as a generalization to both the Camassa–Holm (CH) and modified Camassa–Holm (mCH) equations. We mainly make a detailed presentation on the effects of varying the CH and mCH nonlocal nonlinearities on the non-uniform dependence and Hölder continuity of the solution map. Using a Galerkin-type approximation method, we first establish the local well-posedness result in Sobolev spaces Hs,s>52, with continuous dependence on the initial data. Then we prove that this dependence is sharp by showing that the data-to-solution map is not uniformly continuous, which is based on well-posedness estimates and the method of approximate solutions. Furthermore, we demonstrate that the solution map is Hölder continuous in the HσHσ topology with 0≤σ