On the local well-posedness for some systems of coupled KdV equations

Using the theory developed by Kenig, Ponce, and Vega, we prove that the Hirota–Satsuma system is locally well-posed in Sobolev spaces Hs(R)×Hs(R)Hs(R)×Hs(R) for 3/4−3/4s>−3/4, by establishing new mixed-bilinear estimates involving the two Bourgain-type spaces Xs,b−α− and Xs,b−α+ adapted to ∂t+α−∂x3 and ∂t+α+∂x3 respectively, where |α+|=|α−|≠0|α+|=|α−|≠0.