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

We investigate some well-posedness issues for the initial value problem (IVP) associated with the system {2i∂tu+q∂x2u+iγ∂x3u=F1(u,w)2i∂tw+q∂x2w+iγ∂x3w=F2(u,w), where F1F1 and F2F2 are polynomials of degree 3 involving uu, ww and their derivatives. This system describes the dynamics of two nonlinear short-optical pulse envelopes u(x,t)u(x,t) and w(x,t)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 Hs(R)×Hs(R)Hs(R)×Hs(R), s≥1/4s≥1/4 and global well-posedness result with data in Sobolev spaces Hs(R)×Hs(R)Hs(R)×Hs(R), s>3/5s>3/5.