Rolls and Snaking in a Swift-Hohenberg Equation with Non-smooth Nonlinearity
Abstract
We study rolls and homoclinic snaking in a variation of the one-dimensional Swift-Hohenberg equation, whose standard forms are prototypical order-parameter models for pattern formation in the sciences. Motivated by classes of differential equation models that involve continuous non-smooth low order nonlinear terms, we replace the standard quadratic-cubic nonlinearity by ν|u|α−u3, α∈ [1,2] with ν> 0. In the vicinity of zero, for α<2 this nonlinearity falls outside the scope of classical Taylor expansion and bifurcation analysis. Our partially analytical and partially numerical results highlight that the non-smooth term modifies the criticality of pattern-forming bifurcations and alters the associated branches of solutions. In particular, α∈(1,2) implies subcriticality of roll bifurcations for any ν>0. At α=1 differentiability is lost, which has a strong impact on the bifurcations of sign-changing rolls including the disappearance of homoclinic snaking. Homoclinic snaking thus emerges non-smoothly as α increases from α=1, and persists when retaining an additional, e.g., quadratic term.
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BibTeX
@misc{masur2026rolls,
title = {{Rolls and Snaking in a Swift-Hohenberg Equation with Non-smooth Nonlinearity}},
author = {Lütfiye Masur and Jens D. M. Rademacher},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24355},
eprint = {2609.24355},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24355},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rolls and Snaking in a Swift-Hohenberg Equation with Non-smooth Nonlinearity
AU - Lütfiye Masur
AU - Jens D. M. Rademacher
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24355
UR - https://arxiv.org/abs/2609.24355
AB - We study rolls and homoclinic snaking in a variation of the one-dimensional Swift-Hohenberg equation, whose standard forms are prototypical order-parameter models for pattern formation in the sciences. Motivated by classes of differential equation models that involve continuous non-smooth low order nonlinear terms, we replace the standard quadratic-cubic nonlinearity by ν|u|^α−u^3, α∈ [1,2] with ν> 0. In the vicinity of zero, for α<2 this nonlinearity falls outside the scope of classical Taylor expansion and bifurcation analysis. Our partially analytical and partially numerical results highlight that the non-smooth term modifies the criticality of pattern-forming bifurcations and alters the associated branches of solutions. In particular, α∈(1,2) implies subcriticality of roll bifurcations for any ν>0. At α=1 differentiability is lost, which has a strong impact on the bifurcations of sign-changing rolls including the disappearance of homoclinic snaking. Homoclinic snaking thus emerges non-smoothly as α increases from α=1, and persists when retaining an additional, e.g., quadratic term.
ER -