Coupled nonlinear partial differential equations in physics and biology

物理和生物学中的耦合非线性偏微分方程

• 批准号: 402035-2011
• 负责人: AbouSalem, Walid
• 金额: $ 1.09万
• 依托单位: University of Saskatchewan
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2011
• 资助国家: 加拿大
• 起止时间: 2011-01-01 至 2012-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=487915
• 关键词: Coupled; nonlinear; partial; differential; equations

The purpose of the proposed research is the mathematical analysis of coupled nonlinear partial differential equations (PDEs) that describe fundamental physical and biological processes and that admit nonlinear wave solutions. Examples of such equations are the Faddeev-Skyrme equations in quantum field theory, multi-component nonlinear Schrodinger equations that describe spinor Bose-Einstein condensates, coupled Schrodinger and Korteweg-de Vries equations that describe interacting surface and internal water waves, and coupled reaction-diffusion equations that describe histone dynamics in cells. Such equations have a very rich mathematical structure, and they admit a plethora of interesting solutions, such as topological solitons and nonabelian vortices. However, their rigorous mathematical analysis has received little attention compared to single PDEs. The main ingredients of the analysis are recent advances in the theory of nonlinear PDEs, harmonic analysis and spectral theory. Depending on the problem, topology also plays an important role.

拟议的研究的目的是耦合的非线性偏微分方程（PDE），描述基本的物理和生物过程，并承认非线性波的解决方案的数学分析。此类方程的示例包括量子场论中的Faddeev-Skyrme方程、描述旋量玻色-爱因斯坦凝聚的多分量非线性薛定谔方程、描述相互作用的表面和内部水波的耦合薛定谔和Korteweg-de弗里斯方程以及描述细胞中组蛋白动力学的耦合反应扩散方程。这样的方程有着非常丰富的数学结构，并且有着大量有趣的解，比如拓扑孤子和非交换涡旋。然而，他们的严格的数学分析相比，受到很少的关注单偏微分方程。分析的主要成分是非线性偏微分方程理论，调和分析和谱理论的最新进展。根据问题的不同，拓扑结构也起着重要的作用。