Dissolution and nucleation of jump discontinuities for one-dimensional periodic peridynamic waves
Abstract
We analyze solutions to a one-dimensional periodic linear model for peridynamic waves, focusing on the setting where the natural energy space allows for spatial discontinuities. Exploiting the high-frequency asymptotics of the peridynamic dispersion relation, we construct an explicit solution whose initial displacement has a single jump discontinuity but becomes continuous at every positive time. By time reversibility, this yields a continuous initial displacement whose evolution develops a jump at any prescribed time and is continuous at every other time. We further quantify the formation of this singularity through a sharp Hölder-norm blow-up estimate obtained via a Littlewood-Paley characterization of Hölder spaces. Finally, by superposing suitably weighted solutions, we construct an evolution exhibiting jump discontinuities on a dense set of times.
Related
- Homogeneous nucleation for two-dimensional Kawasaki dynamics
- Weight-adapted ANOVA kernels for high-dimensional periodic approximation
- Sharp non-uniqueness of singular solutions to the one-dimensional periodic cubic nonlinear Schrödinger equation
- Uniform displacement bounds and Gibbs limits for periodic one-dimensional Riesz gases
- Isotropy sets of periodic geodesic flows on three dimensional manifolds
BibTeX
@misc{coclite2026dissolution,
title = {{Dissolution and nucleation of jump discontinuities for one-dimensional periodic peridynamic waves}},
author = {Giuseppe Maria Coclite and Serena Dipierro and Francesco Maddalena and Gianluca Orlando and Enrico Valdinoci},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30041},
eprint = {2609.30041},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30041},
}
RIS
TY - UNPB
TI - Dissolution and nucleation of jump discontinuities for one-dimensional periodic peridynamic waves
AU - Giuseppe Maria Coclite
AU - Serena Dipierro
AU - Francesco Maddalena
AU - Gianluca Orlando
AU - Enrico Valdinoci
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30041
UR - https://arxiv.org/abs/2609.30041
AB - We analyze solutions to a one-dimensional periodic linear model for peridynamic waves, focusing on the setting where the natural energy space allows for spatial discontinuities. Exploiting the high-frequency asymptotics of the peridynamic dispersion relation, we construct an explicit solution whose initial displacement has a single jump discontinuity but becomes continuous at every positive time. By time reversibility, this yields a continuous initial displacement whose evolution develops a jump at any prescribed time and is continuous at every other time. We further quantify the formation of this singularity through a sharp Hölder-norm blow-up estimate obtained via a Littlewood-Paley characterization of Hölder spaces. Finally, by superposing suitably weighted solutions, we construct an evolution exhibiting jump discontinuities on a dense set of times.
ER -