Well-Posedness for a Nonlinear Schrödinger Equation with Quadratic Derivative Nonlinearities for Bounded Primitive Initial Data

We consider the Cauchy problem for a quadratic derivative nonlinear Schrödinger equation whose nonlinearity is a linear combination of ∂x(u2) and ∂x(|u|2). We prove the local well-posedness in the L2-based Sobolev space Hs(R) for s ≥ 0 with bounded primitives. Moreover, we prove the global well-posedness in Hs(R) for s ≥ 1 and a special case of the coefficients of nonlinearities.