| Abstract: |
| We will discuss the local well-posedness for Ablowitz-Ladik (AL) and Discrete Nonlinear Schr\{o}dinger (DNLS) equations supplemented with a broad class of nonzero boundary conditions and, in addition, derive analytical upper bounds for the minimal guaranteed lifespan of their solutions. These bounds suggest finite-time collapse (blow-up) of solutions in the case of the AL systems and the phenomenon of quasi-collapse in the case of DNLS systems. Numerical simulations confirm another theoretical result on the proximity of the dynamics between the two models over time scales up to the common solution lifespan. Finally, for power nonlinearities, we prove the asymptotic equivalence between the two discrete models in the continuous limit. |
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