Dynamical System Approach in Partial Differential Equations

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The main goal of this project is to study dynamical behavior of special solutions for some hyperbolic type partial differential equations (PDEs), which include nonlinear wave equation, nonlinear Schrodinger equation and compressible Euler equation. These equations are widely used to model various problems arising from physics, engineering, biology, finance, etc. The PI aims to construct certain special solutions for those aforementioned equations and their perturbations to understand the modeled phenomena. The perturbations considered in this project have either different scales or randomness, which requires new mathematical ideas and tools to handle. The successfully constructed solutions will not only provide new theoretical results but also stimulate efficient numerical algorithms to compute solutions in multi-scale and stochastic dynamical systems. More specifically, the PI intends to study the following problems rigorously. The first category of problems is to construct special dynamical structures for three kinds of hyperbolic PDEs, which include a system of nonlinear wave equations, the cubic focusing Schrodinger equation and quasilinear Schrodinger equation in high dimensions. All these problems have multiple spatio-temporal scales, which requires new mathematical ideas and analytical tools to handle. The next problem is to study the persistence of Sine-Gordon breather under multiplicative noise perturbation. The main aim is to understand the effect of stochastic forcing. A new method stemming from invariant manifold, random dynamical system and ergodic theory needs to be developed. The last problem in this project is to study the asymptotic behavior of solutions near some steady states for isentropic and non-isentropic compressible flows by using ideas of invariant manifold. The PI will extend existing ideas and develop techniques both on geometric and analytic aspects to provide not only a solution but also a new perspective of this problem.