发表机构
University of Hamburg（汉堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究通过将Swift-Hohenberg方程的非线性替换为非光滑项，发现非光滑性改变分岔临界性，导致roll分岔亚临界，并在$\alpha=1$时破坏同宿snaking，其随$\alpha$增加非光滑地出现。
AI 中文摘要
我们研究一维Swift-Hohenberg方程变体中的rolls和同宿snaking，该方程的标准形式是科学中图案形成的典型序参量模型。受涉及连续非光滑低阶非线性项的微分方程模型类别的启发，我们将标准二次-三次非线性替换为$\nu |u|^\alpha-u^3$，其中$\alpha \in [1,2]$且$\nu > 0$。在零附近，对于$\alpha<2$，该非线性超出经典泰勒展开和分岔分析的范围。我们的部分解析和部分数值结果强调，非光滑项修改了图案形成分岔的临界性，并改变了相关的解分支。特别地，$\alpha\in(1,2)$意味着对于任何$\nu>0$，roll分岔都是亚临界的。在$\alpha=1$时，可微性丧失，这对变号rolls的分岔产生强烈影响，包括同宿snaking的消失。因此，当$\alpha$从$\alpha=1$增加时，同宿snaking以非光滑方式出现，并且在保留额外项（例如二次项）时持续存在。
英文摘要
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 $ν|u|^α-u^3$, $α\in [1,2]$ 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, $α\in(1,2)$ 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.
Comments26 pages, 20 figures
