On the Global Behavior of Solutions of a Coupled System of Nonlinear Schrodinger Equation

We mainly study a system of two coupled nonlinear Schrodinger equations where one equation includes gain and the other one includes losses. This model constitutes a generalization of the model of pulse propagation in birefringent optical fibers. We aim in this study at partially answering a question of some authors in [1]: "Is the H-1-norm of the solution globally bounded in the Manakov case, when. gamma < kappa ?" We found that in the Manakov case, and when gamma < kappa, the solution stays in L-2(0, T; H-1), and also that the H-1-norm of the solution cannot blow up in finite time. In the Manakov case, an estimate of the total energy is provided, which is different from that has been given in [1]. These results are corroborated by numerical results that have been obtained with a finite element solver well adapted for that purpose.