Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2312.03683

The discrete nonlinear Schr\"odinger equation with linear gain and nonlinear loss: the infinite lattice with nonzero boundary conditions and its finite dimensional approximations

Georgios Fotopoulos·Nikos I. Karachalios·Vassilis Koukouloyannis·Paris Kyriazopoulos·Kostas Vetas
View original source ↗Read source document ↗Export metadata ↓

Abstract

The study of nonlinear Schr\"odinger-type 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 Schr\"odinger 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 initial-boundary 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 gain-loss 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 long-time evolution of the system.

Explore related subjects

math-phmath.MPnlin.PS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Georgios Fotopoulos, Nikos I. Karachalios, Vassilis Koukouloyannis, Paris Kyriazopoulos, Kostas Vetas. 2023-12-06. The discrete nonlinear Schr\"odinger equation with linear gain and nonlinear loss: the infinite lattice with nonzero boundary conditions and its finite dimensional approximations. https://arxiv.org/abs/2312.03683

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

KEEP EXPLORING

Related papers

arXiv2026

Why we should condition denoising diffusion generative models on windows of past observations

Data assimilation (DA) is, traditionally, a cycling process that relies on time-dependent priors to propagate information from past observations to future cycles. Using denoising diffusion generative modeling for DA is challenging because standard approaches use a fixed training data set, which in turn leads to a static prior that ignores information from past observations. Because past observations are ignored, DA systems with static priors lead to larger posterior errors than cycling DA systems. Incorporating time-dependent priors into generative models, however, requires expensive and frequent retraining. Motivated by linear systems theory - where the dependence of a prediction of a Kalman filter on past observations decays exponentially - we condition diffusion models on short windows of past observations. Specifically, we describe training procedures for two frameworks: a diffusion DA system predicting the current state given a set of past observations, and a diffusion ``direct observation prediction'' (DOP) system, predicting future observations given a set of past observations. Using a canonical linear system, we show that both systems can achieve the minimal posterior error characteristic of a fully-cycled DA/DOP system, without re-training, provided the time windows are long enough. The linear setup ensures analytical tractability, avoids confounding neural network training errors, and confirms that conditioning on windows of past observations is required for efficient and accurate diffusion-based DA or DOP.

math-ph↗
arXiv2026

The kinematic structures and the inertial geometry of a moving charge

We ask how much of the geometry a charged particle moves in is fixed by its motion, and how much a particle must bring. We ask of a symplectic structure only that it relate velocity to momentum as Hamilton's equations do, and we ask it of every energy at once. In particular, we show that the structures meeting that demand are the canonical one and its twists by a closed two-form of the base. A field provides the two-form, a particle the multiplier before it, which we identify constitutively with its charge. Thus, a single energy governs a family of structures, and each particle takes the one its charge fixes. We then ask what a particle must bring to be given a momentum, and we show that the degree of that map settles the degree at which a field enters Newton's Second Law. An antisymmetric bilinear form returns no Lorentz force, whilst a Randers metric returns one --- a length whose difference from a Riemannian one is linear in the velocity. Moreover, we find that metric already within the twisted structure, as its primitive over a level of the free energy, and its law of transport to be nonlinear, no affine connection being known to serve. Under an indefinite signature the length parts from the dynamics, and the extremals turn from shortest to longest. On the round sphere a monopole flux leaves no such metric, whilst the transport remains and prequantisation, given a unit of action, restricts the charge to a lattice. In this manner, we conclude that each charge-to-mass ratio receives a geometry of its own, so that by a functionalist criterion none of them is the spacetime of a charged particle.

math-ph↗