Coupled nonlinear partial differential equations in physics and biology

提出的研究目的是耦合非线性偏微分方程（PDEs）的数学分析，描述基本的物理和生物过程，并承认非线性波解。这些方程的例子有量子场论中的faddev - skyrme方程，描述自旋玻色-爱因斯坦凝聚体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薛定谔和Korteweg-de Vries方程，以及描述细胞中组蛋白动力学的耦合反应-扩散方程。这样的方程具有非常丰富的数学结构，并且它们承认有大量有趣的解，例如拓扑孤子和非阿贝尔涡旋。然而，与单一偏微分方程相比，他们严谨的数学分析很少受到关注。分析的主要成分是非线性偏微分方程理论、谐波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。

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.