Modeling soliton-potential interactions for the 1-D cubic NLS via an invariant-preserving finite-difference method
Abstract
We consider the 1-D focusing cubic NLS with a smooth, spatially-dependent potential which decays rapidly as x → ± ∞. As is well known, under sufficient regularity conditions, solutions of the cubic NLS with such a spatially-dependent potential conserve both mass and energy. Additionally, in the absence of a potential, the 1-D focusing cubic NLS admits boosted soliton solutions. Furthermore, in the presence of a potential, the interaction of the potential with boosted soliton solutions to the free case is of active interest in the literature, particularly with respect to the dependence of this soliton-potential interaction on the velocity of the incident soliton. We adapt a finite-difference scheme to the setting of a smooth, rapidly-decaying spatially-dependent potential and show that this adaptation preserves the appropriate mass and energy invariants. Additionally, we use this algorithm to model soliton-potential interactions at a variety of incident soliton velocities.
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BibTeX
@misc{jr2026modeling,
title = {{Modeling soliton-potential interactions for the 1-D cubic NLS via an invariant-preserving finite-difference method}},
author = {Anthony D. Stone Jr. and Christopher C. Hogan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17486},
eprint = {2610.17486},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17486},
}
RIS
TY - UNPB
TI - Modeling soliton-potential interactions for the 1-D cubic NLS via an invariant-preserving finite-difference method
AU - Anthony D. Stone Jr.
AU - Christopher C. Hogan
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17486
UR - https://arxiv.org/abs/2610.17486
AB - We consider the 1-D focusing cubic NLS with a smooth, spatially-dependent potential which decays rapidly as x → ± ∞. As is well known, under sufficient regularity conditions, solutions of the cubic NLS with such a spatially-dependent potential conserve both mass and energy. Additionally, in the absence of a potential, the 1-D focusing cubic NLS admits boosted soliton solutions. Furthermore, in the presence of a potential, the interaction of the potential with boosted soliton solutions to the free case is of active interest in the literature, particularly with respect to the dependence of this soliton-potential interaction on the velocity of the incident soliton. We adapt a finite-difference scheme to the setting of a smooth, rapidly-decaying spatially-dependent potential and show that this adaptation preserves the appropriate mass and energy invariants. Additionally, we use this algorithm to model soliton-potential interactions at a variety of incident soliton velocities.
ER -