On the Cauchy problem for a coupled system of third-order nonlinear Schrodinger equations

We investigate some well-posedness issues for the initial value problem (IVP) associated with the system {2i partial derivative(t)u + q partial derivative(2)(x)u + i gamma partial derivative(3)(x)u = F(1)(u, w) 2i partial derivative(t)w + q partial derivative(2)(x)w + i gamma partial derivative(3)(x) = F(2)(u, w), where F(1) and F(2) are polynomials of degree 3 involving u, w and their derivatives. This system describes the dynamics of two nonlinear short-optical pulse envelopes w(x, t) and w(x, t) in fibers (Porsezian et al. (1994) [1] and Hasegawa & Kodama (1987) [2]). We prove sharp local well-posedness result for the IVP with data in Sobolev spaces H(s)(R) x H(s)(R), s >= 1/4 and global well-posedness result with data in Sobolev spaces H(s)(R) x H(s)(R), s > 3/5. (C) 2010 Elsevier Ltd. All rights reserved.