Long-time continuum limits for twisted Schrödinger lattices
Abstract
We prove polynomial-in-time Sobolev bounds on the approximation error between the twisted discrete nonlinear Schrödinger equation and its magnetic continuum limit. The analysis exploits complete integrability of the continuum flow through its Birkhoff representation to control approximation by a non-integrable Hamiltonian finite-difference lattice. We establish a focusing-defocusing dichotomy in uniform polynomial control: the bounds hold for arbitrary defocusing data and focusing data of sufficiently small mass, while modulational instability and aliasing yield an obstruction for large focusing data. The accompanying algebraic spatial rates are generically optimal in the Baire-category sense. The proof combines Birkhoff coordinates with modified energies for discrete Sobolev growth, while Fourier filtering recovers uniform spacetime estimates below the energy space despite degenerate lattice dispersion.
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BibTeX
@misc{choi2026longtime,
title = {{Long-time continuum limits for twisted Schrödinger lattices}},
author = {Brian Choi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34081},
eprint = {2609.34081},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34081},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Long-time continuum limits for twisted Schrödinger lattices
AU - Brian Choi
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34081
UR - https://arxiv.org/abs/2609.34081
AB - We prove polynomial-in-time Sobolev bounds on the approximation error between the twisted discrete nonlinear Schrödinger equation and its magnetic continuum limit. The analysis exploits complete integrability of the continuum flow through its Birkhoff representation to control approximation by a non-integrable Hamiltonian finite-difference lattice. We establish a focusing-defocusing dichotomy in uniform polynomial control: the bounds hold for arbitrary defocusing data and focusing data of sufficiently small mass, while modulational instability and aliasing yield an obstruction for large focusing data. The accompanying algebraic spatial rates are generically optimal in the Baire-category sense. The proof combines Birkhoff coordinates with modified energies for discrete Sobolev growth, while Fourier filtering recovers uniform spacetime estimates below the energy space despite degenerate lattice dispersion.
ER -