We investigate some well-posedness issues for the initial value problem associated to the system for given data in low order Sobolev spaces Hs(R) × Hs(R). We prove local and global well-posedness results utilizing the sharp smoothing es-timates associated to the linear problem combined with the contraction mapping principle. For data with small Sobolev norm we obtain global solution whenever s ≥ 0 by using global smoothing estimates. In particular, for data satisfying where S is solitary wave solution, we get global solution whenever s > 3/4. To prove this last result, we apply the splitting argument introduced by Bourgain [5] and further simplied by Fonseca, Linares and Ponce [6,7].131-2126Ablowitz, M., Kaup, D., Newell, A., Segur, H., Phys. Rev. Lett. (1973) Nonlinear evolution equations of physical significance, 31, pp. 125-127Alarcon, E., Angulo, J., Montenegro, J.F., Nonlinear Analysis (1999) Stability and instability of solitary waves for a nonlinear dispersive system, 36, pp. 1015-1035Angulo, J., Bona, J., Linares, F., Scialom, M., Nonlinearity (2002) Scaling, stability and singularities for nonlinear dispersive wave equations: the critical case, 15, pp. 759-786Bona, J.L., Souganidis, P., Strauss, W., Proc. Roy. Soc. London Ser A (1987) Stability and instability of solitary waves of Korteweg-de Vries type equation, 411, pp. 395-412Bourgain, J., Internat. Math. Res. Notices (1998) Refinements of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity, 5, pp. 253-283Fonseca, G., Linares, F., Ponce, G., Communications in PDE (1999) Global well-posedness for the modified Kortewegde Vries equation, 24, pp. 683-705Fonseca, G., Linares, F., Ponce, G., Proc. Amer. Math. Soc. (2003) Global existence of the critical generalized KdV equation, 131, pp. 1847-1855Friedman, A., Inc (1969) Partial Differential Equations, Holt, Rinehart and WinstonGrillakis, M., Shatah, J., Strauss, W., J. Funct. Anal. (1987) Stability theory of solitary waves in the presence of symmetry I, 74, pp. 160-197Hakkaev, S., Kirchev, K., (2003) Stability of solitary waves for a nonlinear dispersive system in critical case, , PreprintKato, T., Studies in Appl. Math. (1983) On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Mathematics Supplementary Studies, 8, pp. 93-128Kenig, C.E., Ponce, G., Vega, L., J. Amer. Math. Soc. (1991) Well-posedness of the initial value problem for the Korteweg-de Vries equation, 4 (2), pp. 323-347Kenig, C.E., Ponce, G., Vega, L., Indiana University Math. J. (1991) Oscillatory integrals and regularity of dispersive equations, 40 (1), pp. 33-69Kenig, C.E., Ponce, G., Vega, L., Comm. Pure Appl. Math. (1993) Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, 46, pp. 527-620Kenig, C.E., Ruiz, A., Trans. Amer. Math. Soc. (1983) A strong type (2,2) estimate for a maximal operator associated to the Schrödinger equation, 230, pp. 239-246Merle, F., J. Amer. Math. Soc. (2001) Existence of blow-up solutions in the energy space for the critical generalized KdV equation, 14, pp. 555-578Montenegro, J.F., Estudo local, global e estabilidade de ondas solitarias (1995) Sistemas de equações não-lineares, , Ph. D. Thesis, IMPA, Rio de JaneiroPanthee, M., (2004) Properties of solutions to some nonlinear dispersive models, , Ph. D. Thesis, IMPA, Rio de JaneiroStein, E., Weiss, G., (1971) Introduction to Fourier Analysis on Euclidean Spaces, , Prince-ton University PressWeinstein, M., Comm. Pure Appl. Math. (1986) Lyapunov stability of ground states of nonlinear dispersive evolution equations, 39, pp. 51-67