On Stabilization Of Solutions Of Complex Coupled Nonlinear Schrödinger Equations

We study the stabilization of nonchaotic periodic and quasi-periodic solutions of both integrable(α=1)and nonintegrable(α=2/3)of CCNLS equations of the form:\begin{eqnarray*} ip_t +p_{xx} +\frac{1}{2} \sigma (| p |^2 +\alpha | q |^2) p &= & \gamma g_1 (x)\exp (-i\omega_1t)\,,\\[5pt] iq_t +q_{xx} +\frac{1}{2} \sigma (\alpha | p |^2 +| q |^2) q &= & \gamma g_2 (x)\exp (-i\omega_2t)\,, \end{eqnarray*}where subscripts mean partial derivatives,p(x,t)andq(x,t)are the orthogonal components of an electric field in a glass fiber,$i=\sqrt{-1}$, the defocusing(σ=-1)and focusing(σ=1)cases are distinguished byσ;g1(x)andg2(x)are periodic functions inxandγ; andω1andω2are parameters. These solutions do not display sensitive dependence on initial conditions. The stabilization of solutions are studied using a feedback control method and their maximal Lyapunov exponents are calculated. Periodic solutions of this system are important in the study of these coupled equations, since they represent stationary or repeatable behavior.