The Cauchy problem for nonlinear quadratic interactions of the Schrödinger type in one dimensional space

Abstract

In this work, we study the well-posedness of the Cauchy problem associated with the coupled Schrödinger equations with quadratic nonlinearities, which appears modeling problems in nonlinear optics. We obtain the local well-posedness for data in Sobolev spaces with low regularity. To obtain the local theory, we prove new bilinear estimates for the coupling terms of the system in the continuous case. Concerning global results, in the continuous case, we establish the global well-posedness in Hs(R)×Hs(R), for some negatives indexes s. The proof of our global result uses the I-method introduced by Colliander et al.