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) x H-s(R) for 3/4 s <= 1. We introduce some Bourgain-type spaces X-s,b(a) for a not equal 0, s, b is an element of R to obtain local well-posedness for the Gear-Grimshaw system in H-s (R) x H-s (R) for s > -3/4, by establishing new mixed-bilinear estimates involving the two Bourgain-type spaces X-s,b(-alpha-) and X-s,b(-alpha+) adapted to partial derivative(t) + alpha(-)partial derivative(3)(x) and alpha + partial derivative(3)(x) respectively, where vertical bar alpha+vertical bar=vertical bar alpha(-)vertical bar not equal 0. (C) 2007 Elsevier Ltd. All rights reserved.692692715