Sobre o problema de Cauchy para interações não lineares do tipo Schrödinger

Abstract

We study the Cauchy problem associated to the coupled Schr¨odinger equations, which appears modeling problems in nonlinear optics, where the initial data are considered in the classical Sobolev spaces, (u0; v0) ϵ Hk (R) x Hs(R). Well-posedness results for this system, in the periodic case, were obtained by Ângulo and Linares in [1]. In this work we develop a local theory for the system, where the regularity (k; s) of the initial data depends on the different situations of the parameter σ > 0. Also, we obtain global well-posedness results when σ ≠ 2 and for negative indices k = s < 0 included in the local theory developed. Finally, we show some ill-posedness results.