Multiprecision computation of bright and dark solitons in the discrete nonlinear Schrödinger equation
Abstract
We study the spectral stability of bright and dark solitons in the discrete nonlinear Schrödinger (DNLS) equation using multiprecision arithmetic. The eigenvalues governing stability are exponentially small in the lattice spacing and cannot be resolved with standard double precision. To address this, we develop a computational framework combining multiprecision arithmetic, an exact Jacobian for the stationary problem, and a squared-operator formulation for spectral analysis. This enables accurate resolution of exponentially small eigenvalues and direct comparison with exponential-asymptotic predictions. Our results show that onsite bright solitons are spectrally stable, whereas intersite bright solitons and both onsite and intersite dark solitons are unstable. Bright solitons require only a few eigenvalues and allow efficient large-scale computations, while dark solitons demand higher precision due to their proximity to the continuous spectrum. Simulations up to N=65,250 grid points (31.7 GB RAM) highlight the necessity of multiprecision arithmetic for capturing beyond-all-orders spectral effects.
Versions
Related
- Comment on "Space-time shifted solitons and solution interactions for a generalized nonlocal nonlinear Schrödinger equation"
- Self-similar vector solitons for the coupled higher-order nonlinear Schrodinger equations in inhomogeneous optical fibers
- Solitons in the nonlinear Schrödinger equation with the pseudo-Raman term, anti-diffusion gain and nonlinear loss
- Stability of Solitons in the Parametrically Driven and Damped Nonlinear Schrödinger Equation in Arbitrary Dimension
- The focusing Calogero--Moser derivative nonlinear Schrödinger equation in the Hardy--Zhidkov space: explicit flow formula and dark multi-solitons
BibTeX
@misc{kusdiantara2026multiprecision,
title = {{Multiprecision computation of bright and dark solitons in the discrete nonlinear Schrödinger equation}},
author = {Rudy Kusdiantara and Farrell T. Adriano and Hadi Susanto},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27663},
eprint = {2609.27663},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27663},
}
RIS
TY - UNPB
TI - Multiprecision computation of bright and dark solitons in the discrete nonlinear Schrödinger equation
AU - Rudy Kusdiantara
AU - Farrell T. Adriano
AU - Hadi Susanto
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27663
UR - https://arxiv.org/abs/2609.27663
AB - We study the spectral stability of bright and dark solitons in the discrete nonlinear Schrödinger (DNLS) equation using multiprecision arithmetic. The eigenvalues governing stability are exponentially small in the lattice spacing and cannot be resolved with standard double precision. To address this, we develop a computational framework combining multiprecision arithmetic, an exact Jacobian for the stationary problem, and a squared-operator formulation for spectral analysis. This enables accurate resolution of exponentially small eigenvalues and direct comparison with exponential-asymptotic predictions. Our results show that onsite bright solitons are spectrally stable, whereas intersite bright solitons and both onsite and intersite dark solitons are unstable. Bright solitons require only a few eigenvalues and allow efficient large-scale computations, while dark solitons demand higher precision due to their proximity to the continuous spectrum. Simulations up to N=65,250 grid points (31.7 GB RAM) highlight the necessity of multiprecision arithmetic for capturing beyond-all-orders spectral effects.
ER -