Slow-moving pattern interfaces in general directions for a two-dimensional Swift-Hohenberg-type equation
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917399090954240 |
|---|---|
| author | Hilder, Bastian Jansen, Jonas |
| author_facet | Hilder, Bastian Jansen, Jonas |
| contents | We rigorously prove the bifurcation of slow-moving pattern interfaces with general direction in a two-dimensional Swift-Hohenberg-type model close to a Turing instability for a large class of nonlinearities. These interfaces describe the invasion of stripe and hexagonal patterns into the spatially homogeneous state and model a possible mechanism for pattern formation, as observed in a wide range of real-world applications. For this, we develop a rigorous framework to establish the existence of such solutions using spatial dynamics and non-standard centre manifold theory. Our approach exploits geometric and algebraic structures generic to $\mathrm{O}(2)$-symmetric pattern-forming systems near a Turing instability, and addresses fundamental technical challenges due to a non-uniform spectral gap around the imaginary axis, quadratic resonances induced by the hexagonal structure, and the high-dimensional phase space of the reduced equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09530 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Slow-moving pattern interfaces in general directions for a two-dimensional Swift-Hohenberg-type equation Hilder, Bastian Jansen, Jonas Analysis of PDEs Pattern Formation and Solitons 35B36, 35B32, 37L10, 34C37, 35Q56, 35B10, 34E15, 35K58 We rigorously prove the bifurcation of slow-moving pattern interfaces with general direction in a two-dimensional Swift-Hohenberg-type model close to a Turing instability for a large class of nonlinearities. These interfaces describe the invasion of stripe and hexagonal patterns into the spatially homogeneous state and model a possible mechanism for pattern formation, as observed in a wide range of real-world applications. For this, we develop a rigorous framework to establish the existence of such solutions using spatial dynamics and non-standard centre manifold theory. Our approach exploits geometric and algebraic structures generic to $\mathrm{O}(2)$-symmetric pattern-forming systems near a Turing instability, and addresses fundamental technical challenges due to a non-uniform spectral gap around the imaginary axis, quadratic resonances induced by the hexagonal structure, and the high-dimensional phase space of the reduced equations. |
| title | Slow-moving pattern interfaces in general directions for a two-dimensional Swift-Hohenberg-type equation |
| topic | Analysis of PDEs Pattern Formation and Solitons 35B36, 35B32, 37L10, 34C37, 35Q56, 35B10, 34E15, 35K58 |
| url | https://arxiv.org/abs/2604.09530 |