Sharp Global Well-Posedness for the Cubic Nonlinear Schrödinger Equation with Third Order Dispersion

We consider the initial value problem (IVP) associated to the cubic nonlinear Schr & ouml;dinger equation with third-order dispersion partial derivative(t)u + i alpha partial derivative(2)(x)u - partial derivative(3)(x)u + i beta|u|(2)u = 0, x, t is an element of R, for given data in the Sobolev space Hs(R). This IVP is known to be locally well-posed for given data with Sobolev regularity s > -1/4 and globally well-posed for s >= 0 (Carvajal in Electron J Differ Equ 2004:1-10, 2004). For given data in H-s(R), 0 > s > -1/4 no global well-posedness result is known. In this work, we derive an almost conserved quantity for such data and obtain a sharp global well-posedness result. Our result answers the question left open in (Carvajal in Electron J Differ Equ 2004:1-10, 2004).