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The study of nonlinear Schr\"odinger-type equations with nonzero boundaryconditions define challenging problems both for the continuous (partialdifferential equation) or the discrete (lattice) counterparts. They areassociated with fascinating dynamics emerging by the ubiquitous phenomenon ofmodulation instability. In this work, we consider the discrete nonlinearSchr\"odinger equation with linear gain and nonlinear loss. For the infinitelattice supplemented with nonzero boundary conditions which describe solutionsdecaying on the top of a finite background, we give a rigorous proof that forthe corresponding initial-boundary value problem, solutions exist for anyinitial condition, if and only if, the amplitude of the background has aprecise value $A_*$ defined by the gain-loss parameters. We argue that thisessential property of this infinite lattice can't be captured by finite latticeapproximations of the problem. Commonly, such approximations are defined bylattices with periodic boundary conditions or as it is shown herein, by amodified problem closed with Dirichlet boundary conditions. For the finitedimensional dynamical system defined by the periodic lattice, the dynamics forall initial conditions are captured by a global attractor. Analytical argumentscorroborated by numerical simulations show that the global attractor istrivial, defined by a plane wave of amplitude $A_*$. Thus, any instabilityeffects or localized phenomena simulated by the finite system can be onlytransient prior the convergence to this trivial attractor. Aiming to simulatethe dynamics of the infinite lattice as accurately as possible, we study thedynamics of localized initial conditions on the constant background andinvestigate the potential impact of the global asymptotic stability of thebackground with amplitude $A_*$ in the long-time evolution of the system.

The discrete nonlinear Schr\"odinger equation with linear gain and  nonlinear loss: the infinite lattice with nonzero boundary conditions and its  finite dimensional approximations