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

Abstract

We investigate some well-posedness issues for the initial value problem (IVP) associated to the system \begin{equation} \{ \begin{array} [c]{l} 2i\partial_{t}u+q\partial_{x}^{2}u+iγ\partial_{x}^{3}u=F_{1}(u,w)\\ 2i\partial_{t}w+q\partial_{x}^{2}w+iγ\partial_{x}^{3}w=F_{2}(u,w), \end{array} . \end{equation} 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 pulses envelopes $u(x,t)$ and $w(x,t)$ in fibers (\cite{31}, \cite{14}). We prove periodic local well-posedness for the IVP with data in Sobolev spaces $H^s(\mathbb{T})\times H^s(\mathbb{T})$, $s\ge 1/2$ and global well-posedness result in Sobolev spaces $H^1(\mathbb{T})\times H^1(\mathbb{T})$.

21 pages