Series approximation of discrete-soliton solutions of the DNLS equation
Abstract
We study standing waves of the discrete nonlinear Schrödinger (DNLS) equation by means of perturbative series expansions in the inter-site coupling parameter ε. Power-series approximations are derived for three representative configurations: a single-site soliton and the two two-site ones, namely the in-phase and the out-of-phase configurations. Although higher-order truncations are increasingly accurate for small ε, the expansions possess a small radius of convergence and cease to be useful well before ε=cal O(1). To extend their range of validity we apply the Euler convergence-acceleration method, constructing an optimally transformed series for each lattice site individually. The transformed solution is shown to have substantially better convergence radius and preserves the localization of the solution much better than the original series solution. We then insert the series profiles into the linearization matrix of the DNLS equation in order to obtain improved approximations of the stability eigenvalues, with particular attention to the unstable real eigenvalue of the in-phase configuration. Optimal truncation of the corresponding expansion yields a closed-form cal O(ε5/2) estimate, which significantly extends the known cal O(ε1/2) leading-order result to considerably larger couplings, while the semi-numerical use of the accelerated profiles reproduces the numerically computed eigenvalue to values of ε of cal O(1). Throughout, both the solution profiles and their stability eigenvalues are benchmarked against the numerical solution obtained through a continuation procedure from the anti-continuum limit.
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BibTeX
@misc{tsiantou2026series,
title = {{Series approximation of discrete-soliton solutions of the DNLS equation}},
author = {V. Tsiantou and V. Koukouloyannis and K. D. Housiadas and P. G. Kevrekidis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38461},
eprint = {2610.38461},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38461},
}
RIS
TY - UNPB
TI - Series approximation of discrete-soliton solutions of the DNLS equation
AU - V. Tsiantou
AU - V. Koukouloyannis
AU - K. D. Housiadas
AU - P. G. Kevrekidis
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38461
UR - https://arxiv.org/abs/2610.38461
AB - We study standing waves of the discrete nonlinear Schrödinger (DNLS) equation by means of perturbative series expansions in the inter-site coupling parameter ε. Power-series approximations are derived for three representative configurations: a single-site soliton and the two two-site ones, namely the in-phase and the out-of-phase configurations. Although higher-order truncations are increasingly accurate for small ε, the expansions possess a small radius of convergence and cease to be useful well before ε=cal O(1). To extend their range of validity we apply the Euler convergence-acceleration method, constructing an optimally transformed series for each lattice site individually. The transformed solution is shown to have substantially better convergence radius and preserves the localization of the solution much better than the original series solution. We then insert the series profiles into the linearization matrix of the DNLS equation in order to obtain improved approximations of the stability eigenvalues, with particular attention to the unstable real eigenvalue of the in-phase configuration. Optimal truncation of the corresponding expansion yields a closed-form cal O(ε^5/2) estimate, which significantly extends the known cal O(ε^1/2) leading-order result to considerably larger couplings, while the semi-numerical use of the accelerated profiles reproduces the numerically computed eigenvalue to values of ε of cal O(1). Throughout, both the solution profiles and their stability eigenvalues are benchmarked against the numerical solution obtained through a continuation procedure from the anti-continuum limit.
ER -