Data dependence for the amplitude equation of surface waves

We consider the amplitude equation for nonlinear surface wave solutions of hyperbolic conservation laws. This is an asymptotic nonlocal, Hamiltonian evolution equation with quadratic nonlinearity. For example, this equation describes the propagation of nonlinear Rayleigh waves, surface waves on current-vortex sheets in incompressible MHD and on the incompressible plasma-vacuum interface. The local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation in noncanonical variables was shown in a previous article. In the present paper we prove the continuous dependence in strong norm of solutions on the initial data. This completes the proof of the well-posedness of the problem in the classical sense of Hadamard