The Cauchy problem for the Fokas–Olver–Rosenau–Qiao equation

It is shown that the initial value problem for the Fokas–Olver–Rosenau–Qiao equation (FORQ) is well-posed in Sobolev spaces HsHs, s>5/2s>5/2, in the sense of Hadamard. Furthermore, it is proved that the dependence on initial data is sharp, i.e. the data-to-solution map is continuous but not uniformly continuous. Also, peakon travelling wave solutions are derived on both the circle and the line and are used to prove that the solution map is not uniformly continuous in HsHs for s<3/2s<3/2.