On the global behaviour of solutions of a coupled system of nonlinear Schrödinger equations

Abstract : We mainly study a system of two coupled nonlinear Schrödinger equations where one equation includes gain and the other one includes losses. This model constitutes a generalization of the model of pulse propagation in bire-fringent optical fibers. We answered partially to a question of some authors in [8], that in the Manakov case, the solution stays in L 2 (0, T ; H 1), that means that the solution can not blow up in finite time. More precisely, the bound that is provided in this paper does not seem to be optimal but different than those that has been given from a previous study [8]. Thanks to the way we treat the a priori estimate, we obtain a sharp bound in L 2 (0, T ; H 1), which would be difficult to reach from the study of other authors [8]. The result is illustrated by numerical results which have been obtained with a finite element solver well adapted for that purpose.
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E Destyl, Silvère Paul Nuiro, Pascal Poullet. On the global behaviour of solutions of a coupled system of nonlinear Schrödinger equations. Studies in Applied Mathematics, Wiley-Blackwell, 2017, 138 (2), pp.227-244. ⟨hal-01270459⟩

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