Interaction Equations for Short and Long Dispersive Waves

We show the time-local well-posedness for a system of nonlinear dispersive equations for the water wave interaction[formula]It is shown that for any initial data (u0, v0)∈Hs(R)×Hs−1/2(R) (s⩾0), the solution for the above equation uniquely exists in a subset ofC((−T, T);Hs)×C((−T, T);Hs−1/2) and depends continuously on the data. By virtue of a special structure of the nonlinear coupling, the solution is stable under a singular limiting process.