Global existence and compact attractors for the discrete nonlinear Schrödinger equation

Abstract

We study the asymptotic behavior of solutions of discrete nonlinear Schrödinger-type (DNLS) equations. For a conservative system, we consider the global in time solvability and the question of existence of standing wave solutions. Similarities and differences with the continuous counterpart (NLS-partial differential equation) are pointed out. For a dissipative system we prove existence of a global attractor and its stability under finite-dimensional approximations. Similar questions are treated in a weighted phase space. Finally, we propose possible extensions for various types of DNLS equations.

Global Existence in Infinite Lattices of Nonlinear Oscillators: The Discrete Klein-Gordon Equation

“…Similar behaviour is shown for the the solutions of the weakly damped DNLS systems [19]. In the unforced case, it can be easily seen that solutions of the weakly damped DNLS systems decay exponentially.…”

“…As a continuation of our previous works on lattice differential equations [18,19], here we consider the following discrete Klein-Gordon equation DKG, considered in higher dimensional lattices (n = (n 1 , n 2 , . .…”

“…Our study takes into account the following reported observations and results: (a) the strong relation between DNLS and DKG systems, through the analysis of slow temporal variations (b) the global existence of solutions, for the conservative or the damped DKG system for β = 1 [24, 25, 33, 37, 38] (c) the global existence results and decay estimates for small solutions of [33], in the case β = ± 1, (d) the unconditional global existence of solutions for the conservative or damped DNLS system [5,19,33].…”

“…The comment (for the proofs we refer to [18,19]), is related with the appearance of energy thresholds, for the existence of such type of solutions for the DKG equation, [33,35].…”

Discrete Schrödinger equations and dissipative dynamical systems

“…We would like to refer to [19] and [5] where the authors consider some discretization in space of the Laplace operator on a finite interval with finite differences and then study the dynamical systems provided by the associated ODE. On the other hand, we would like also to point out the study of discrete nonlinear Schrödinger equations as an infinite-dimensional dynamical system in a lattice in [13]; the authors consider the ODE in the infinite dimensional space l 2 (Z) defined at each point j ∈ Z by (u j ) t + γu j + i(2u j − u j+1 − u j−1 ) + i|u j | 2 u j = f j .…”