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In this work I study the well-posedness of the Cauchy problem associated withthe coupled Schr\"odinger equations {with quadratic nonlinearities}, whichappears modeling problems in nonlinear optics. I obtain the localwell-posedness for data {in Sobolev spaces} with low regularity. {To obtain}the local theory, I prove new bilinear estimates for the coupling terms of thesystem in the continuous case. Concerning global results, in the continuouscase, I establish the global well-posedness in $H^s(\mathbb{R})\timesH^s(\mathbb{R})$, for some negatives indexes $s$. The proof of the globalresult uses the \textbf{I}-method introduced by Colliander, Keel, Staffilani,Takaoka and Tao.

The Cauchy Problem for nonlinear Quadratic Interactions of the Schr\"odinger type in one dimensional space