A machine-learned dynamical phase diagram of the one-dimensional nonlinear Schrödinger equation with quasiperiodic disorder and a static field
Abstract
We map the transport regimes of a wave packet evolving under the one-dimensional discrete nonlinear Schrödinger (Gross-Pitaevskii) equation with a quasiperiodic Aubry-André potential and a static (Stark) field, as a function of self-interaction g, field F, and quasiperiodic strength λ, for two initial states (delta and Gaussian). Instead of one expensive long-time simulation per grid point, we run 3600 short simulations, reduce each to eight dynamical features, and cluster parameter space with a Gaussian mixture model. Clustering finds the boundaries between regimes unsupervised; each cluster's name is then assigned by a hand-calibrated rule, and an ablation shows clustering measurably smooths those boundaries rather than merely relabeling them. The five resulting regimes (ballistic, localized, oscillatory-localized, subdiffusive, and self-trapped) are validated against 100 long runs on a lattice eight times larger than the short sweep, so the validation window is longer than the training window even for the fastest-spreading regime. The short-time clusters predict the long-time asymptotic spreading exponent (α∞ = 2.01±0.07 ballistic, 0.28±0.16 subdiffusive, consistent with the weak-chaos prediction α=1/3) and, independently, a long-time retention Π0=0.72±0.32 for the self-trapped cluster, whose exponent alone is not diagnostic. The initial-state contrast is striking: the delta state develops a broad self-trapping wedge (onset at g=3.907±0.017 at λ=0) that invades the ballistic and localized regions as g grows, while the Gaussian state shows no self-trapping, instead opening a subdiffusive corridor along the Aubry-André critical line that widens with g. The scheme yields a full 15×15 phase map in minutes per slice, where direct asymptotic simulation (t≳106 per point) is prohibitive.
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BibTeX
@misc{perez2026machinelearned,
title = {{A machine-learned dynamical phase diagram of the one-dimensional nonlinear Schrödinger equation with quasiperiodic disorder and a static field}},
author = {Marcos Pérez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10045},
eprint = {2609.10045},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10045},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A machine-learned dynamical phase diagram of the one-dimensional nonlinear Schrödinger equation with quasiperiodic disorder and a static field
AU - Marcos Pérez
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.10045
UR - https://arxiv.org/abs/2609.10045
AB - We map the transport regimes of a wave packet evolving under the one-dimensional discrete nonlinear Schrödinger (Gross-Pitaevskii) equation with a quasiperiodic Aubry-André potential and a static (Stark) field, as a function of self-interaction g, field F, and quasiperiodic strength λ, for two initial states (delta and Gaussian). Instead of one expensive long-time simulation per grid point, we run 3600 short simulations, reduce each to eight dynamical features, and cluster parameter space with a Gaussian mixture model. Clustering finds the boundaries between regimes unsupervised; each cluster's name is then assigned by a hand-calibrated rule, and an ablation shows clustering measurably smooths those boundaries rather than merely relabeling them. The five resulting regimes (ballistic, localized, oscillatory-localized, subdiffusive, and self-trapped) are validated against 100 long runs on a lattice eight times larger than the short sweep, so the validation window is longer than the training window even for the fastest-spreading regime. The short-time clusters predict the long-time asymptotic spreading exponent (α_∞ = 2.01±0.07 ballistic, 0.28±0.16 subdiffusive, consistent with the weak-chaos prediction α=1/3) and, independently, a long-time retention Π_0=0.72±0.32 for the self-trapped cluster, whose exponent alone is not diagnostic. The initial-state contrast is striking: the delta state develops a broad self-trapping wedge (onset at g=3.907±0.017 at λ=0) that invades the ballistic and localized regions as g grows, while the Gaussian state shows no self-trapping, instead opening a subdiffusive corridor along the Aubry-André critical line that widens with g. The scheme yields a full 15×15 phase map in minutes per slice, where direct asymptotic simulation (t≳10^6 per point) is prohibitive.
ER -