Lower bounds on the radius of analyticity for a system of nonlinear quadratic interactions of the Schrödinger-type equations

In this paper, we study the Cauchy problem for a system of nonlinear Schrödinger equations with quadratic interactions and initial data belonging to a class of analytic Gevrey functions. Here, we present a local well-posedness result in the analytic Gevrey class \(G^{\sigma ,s}\times G^{\sigma ,s}\) $G^{\sigma ,s}\times G^{\sigma ,s}$ by proving some bilinear estimates in Bourgain’s space with exponential weight. Furthermore, we prove that the obtained solution can be extended to any time \(T>0\) $T>0$ , as long as the radius of the spatial analyticity \(\sigma \) $\sigma$ is bounded below by \(cT^{-2}\) $c{T}^{-2}$ , if \(0<a <1/2\) $0<a<1/2$ , or \(cT^{- 4}\) $c{T}^{-4}$ , if \(a>1/2\) $a>1/2$ .