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

Abstract

Using the theory developed by Kenig, Ponce, and Vega, we prove that the Hirota–Satsuma system is locally well-posed in Sobolev spaces H s ( R ) × H s ( R ) for 3 / 4 < s ≤ 1 . We introduce some Bourgain-type spaces X s , b a for a ≠ 0 , s , b ∈ R to obtain local well-posedness for the Gear–Grimshaw system in H s ( R ) × H s ( R ) for s > − 3 / 4 , by establishing new mixed-bilinear estimates involving the two Bourgain-type spaces X s , b − α − and X s , b − α + adapted to ∂ t + α − ∂ x 3 and ∂ t + α + ∂ x 3 respectively, where | α + | = | α − | ≠ 0 .