Local and global well-posedness for a quadratic Schrodinger system on Zoll manifolds

We consider the initial value problem (IVP) associated to a quadratic Schriidinger system {i partial derivative(t)upsilon +/- Delta(g)upsilon - upsilon = epsilon(1)u (upsilon) over bar, t is an element of R, x is an element of M, i sigma partial derivative(t)u +/- Delta(g)u - alpha u - epsilon(2)/2 upsilon(2), sigma < 0, alpha is an element of R, epsilon(i) is an element of C (i = 1, 2), (upsilon(0), u(0)) = (upsilon(0), u(0)), posed on a d-dimensional compact Zoll manifold M. Considering sigma = theta/beta with theta, beta is an element of {n(2) : n is an element of Z} we derive a bilinear Strichartz type estimate and use it to prove the local well-posedness results for given data (upsilon(0), u(0)) is an element of H-s (M) x H-s (M) whenever s > 1/4 when d = 2 and s > d-2/2 when d >= 3. Moreover, in dimensions 2 and 3, we use a Gagliardo-Nirenberg type inequality and conservation laws to prove that the local solution can be extended globally in time whenever s >= 1. (C) 2020 Elsevier Inc. All rights reserved.