The discrete nonlinear Schrödinger equation with linear gain and nonlinear loss: the infinite lattice with nonzero boundary conditions and its finite dimensional approximations
Abstract
The study of nonlinear Schrödingertype equations with nonzero boundary conditions define challenging problems both for the continuous (partial differential equation) or the discrete (lattice) counterparts. They are associated with fascinating dynamics emerging by the ubiquitous phenomenon of modulation instability. In this work, we consider the discrete nonlinear Schrödinger equation with linear gain and nonlinear loss. For the infinite lattice supplemented with nonzero boundary conditions which describe solutions decaying on the top of a finite background, we give a rigorous proof that for the corresponding initialboundary value problem, solutions exist for any initial condition, if and only if, the amplitude of the background has a precise value $A_*$ defined by the gainloss parameters. We argue that this essential property of this infinite lattice can't be captured by finite lattice approximations of the problem. Commonly, such approximations are defined by lattices with periodic boundary conditions or as it is shown herein, by a modified problem closed with Dirichlet boundary conditions. For the finite dimensional dynamical system defined by the periodic lattice, the dynamics for all initial conditions are captured by a global attractor. Analytical arguments corroborated by numerical simulations show that the global attractor is trivial, defined by a plane wave of amplitude $A_*$. Thus, any instability effects or localized phenomena simulated by the finite system can be only transient prior the convergence to this trivial attractor. Aiming to simulate the dynamics of the infinite lattice as accurately as possible, we study the dynamics of localized initial conditions on the constant background and investigate the potential impact of the global asymptotic stability of the background with amplitude $A_*$ in the longtime evolution of the system.
 Publication:

arXiv eprints
 Pub Date:
 December 2023
 DOI:
 10.48550/arXiv.2312.03683
 arXiv:
 arXiv:2312.03683
 Bibcode:
 2023arXiv231203683F
 Keywords:

 Mathematical Physics;
 Nonlinear Sciences  Pattern Formation and Solitons;
 35Q55;
 37L60
 EPrint:
 23 pages, 10 figures