Local well‐posedness for a system of quadratic nonlinear Schrödinger equations in one or two dimensions

Abstract

In this article, the local well‐posedness of Cauchy's problem is explored for a system of quadratic nonlinear Schrödinger equations in the space Lp(Rn). In a special case of mass resonant 2 × 2 system, it is well known that this problem is well posed in Hs(s≥0) and ill posed in Hs(s < 0) in two‐space dimensions. By translation on a linear semigroup, we show that the general system becomes locally well posed in Lp(R2) for 1 < p < 2, for which p can arbitrarily be close to the scaling limit pc=1. In one‐dimensional case, we show that the problem is locally well posed in L1(R); moreover, it has a measure valued solution if the initial data are a Dirac function. Copyright © 2016 John Wiley & Sons, Ltd.

On a system of Schrödinger equations with general quadratic-type nonlinearities

“…Inspired by these works we intent to provide sufficient conditions on the interactions terms, f k , to study the dynamics of system (1.1). General nonlinearities with quadratic interactions were considered for example in references [26], [25] and [40]. These works were dedicated to the study the Cauchy problem in two dimensions.…”

“…Here we consider system (1.1) in dimensions 1 ≤ n ≤ 6. Also, it is important to mention that our nonlinearities include the ones considered in [26] and [40]. However, in our work no explicit form is assumed on the interaction terms.…”