Approximately invariant manifolds and global dynamics of spike states

• 成果类型: Article
• 署名作者: Bates, Peter W.; Lu, Kening; Zeng, Chongchun
• 署名单位: Michigan State University; Brigham Young University; University System of Georgia; Georgia Institute of Technology
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-008-0141-y
• 发表日期: 2008
• 页码: 355-433
• 关键词: nonlinear schrodinger-equations; boundary peak solutions; least-energy solutions; metastable patterns; layer solutions; asymptotic stability; stationary solutions; multipeak solutions; positive solutions; Neumann problem

摘要

We investigate the existence of a true invariant manifold given an approximately invariant manifold for an infinite-dimensional dynamical system. We prove that if the given manifold is approximately invariant and approximately normally hyperbolic, then the dynamical system has a true invariant manifold nearby. We apply this result to reveal the global dynamics of boundary spike states for the generalized Allen-Cahn equation.