Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0503298

Global Existence and Compact Attractors for the Discrete Nonlinear Schrödinger equation

Nikos I. Karachalios·Athanasios N. Yannacopoulos
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.CAmath.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Nikos I. Karachalios, Athanasios N. Yannacopoulos. 2005-03-15. Global Existence and Compact Attractors for the Discrete Nonlinear Schrödinger equation. https://arxiv.org/abs/math/0503298

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Regular specular differentiation in Euclidean spaces

We study the regular specular derivative, a generalized derivative defined at every point where both one-sided derivatives exist and are finite. Geometrically, it is the slope of the mirror that reflects the left tangent ray into the right one. In one variable we derive computational formulas, prove inverse function and rotation rules, establish Quasi-Rolle's Theorem and the Quasi-Mean Value Theorem, and obtain a derivative-limit theorem, which shows that twice regularly specularly differentiable functions are continuously differentiable. We also prove both parts of the Fundamental Theorem of Calculus. In several variables we introduce specular gradients, directional derivatives, tangent hyperplanes, and normal vectors, show that a continuous specular gradient forces classical differentiability, and characterize when the specular tangent hyperplane is unique.

math.CA↗
arXiv2026

Prevalent smoothness in inhomogeneous Besov spaces

In this article, we prove that, under some assumptions on the so-called environment, prevalent functions in inhomogeneous Besov spaces recently introduced by Barral-Seuret in 2023 are multifractal, with a singularity spectrum that we determine. This completes the previous Baire generic results already obtained.

math.CA↗
arXiv2026

Lebesgue Covering Theorem and level sets of continuous functions

We formulate and prove a dimension-theoretic generalization of a version of the Lebesgue Covering Theorem. A generalized $n$-dimensional version of the Steinhaus Chessboard Theorem, recently proved algorithmically by Turzański and Ziajor, is a particular case of this result. Moreover, we study two types of sets associated with a continuous function $g \colon [0,1]^n \to \mathbb{R}$. Namely, the set of all points $p \in \mathbb{R}$ such that the fiber $g^{-1}[\left\{p\right\}]$ connects $i$th opposite faces of $[0,1]^n$, and the set of all points $p \in \mathbb{R}$ such that the fiber $g^{-1}[\left\{p\right\}]$ separates $i$th opposite faces of $[0, 1]^n$.

math.CA↗