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We discuss the finite-time collapse, also referred as blow-up, of thesolutions of a discrete nonlinear Schr\"{o}dinger (DNLS) equation incorporatinglinear and nonlinear gain and loss. This DNLS system appears in many inherentlydiscrete physical contexts as a more realistic generalization of theHamiltonian DNLS lattice. By using energy arguments in finite and infinitedimensional phase spaces (as guided by the boundary conditions imposed), weprove analytical upper and lower bounds for the collapse time, valid for boththe defocusing and focusing cases of the model. In addition, the existence of acritical value in the linear loss parameter is underlined, separating finitetime-collapse from energy decay. The numerical simulations, performed for awide class of initial data, not only verified the validity of our bounds, butalso revealed that the analytical bounds can be useful in identifying twodistinct types of collapse dynamics, namely, extended or localized. Pending onthe discreteness /amplitude regime, the system exhibits either type of collapseand the actual blow-up times approach, and in many cases are in excellentagreement, with the upper or the lower bound respectively. When these times liebetween the analytical bounds, they are associated with a nontrivial mixing ofthe above major types of collapse dynamics, due to the corroboration ofdefocusing/focusing effects and energy gain/loss, in the presence ofdiscreteness and nonlinearity.

Collapse dynamics for the discrete nonlinear Schr\"odinger equation with gain and loss