The existence of a global attractor for the discrete nonlinear Schrödinger equation. II. Compactness without tail estimates in $\mathbb{Z}^N$, $N\geq1$, lattices

Abstract

Recent studies of lattice dynamics, with respect to the existence of global compact attractors for certain discrete evolution equations, are based on the derivation of ‘tail estimates of the solution’. We show that, due to the specific nature of the discrete nonlinear Schrödinger (DNLS) system which gives rise to a simple energy equation, the method developed by J. M. Ball is applicable, and provides an alternative proof on the existence of the compactness of the attractor for the weakly damped and driven DNLS equation considered in $\mathbb{Z}^N$, $N\geq1$, lattices, without the usage of tail estimates. The approach covers various DNLS-type equations of physical significance.