Coupled nonlinear partial differential equations in physics and biology

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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.