We consider a chain of coupled pendula pairs, where each pendulum is connected to the nearest neighbors in the longitudinal and transverse directions. The common strings in each pair are modulated periodically by an external force. In the limit of small coupling and near the 1 : 2 parametric resonance, we derive a novel system of coupled PT -symmetric discrete nonlinear Schrödinger equation, which has Hamiltonian symmetry but has no gauge symmetry. By using the conserved energy, we find the parameter range for the linear and nonlinear stability of the zero equilibrium. Numerical experiments illustrate how destabilization of the zero equilibrium takes place when the stability constraints are not satisfied. The central pendulum excites nearest pendula and this process continues until a dynamical equilibrium is reached where each pendulum in the chain oscillates at a finite amplitude.
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 birefringent optical fibers. We aim in this study at partially answering a question of some authors in [1]: “Is the H1‐norm of the solution globally bounded in the Manakov case, when γ<κ ?” We found that in the Manakov case, and when γ<κ, the solution stays in L2false(0,T;H1false), and also that the H1‐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.
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