Dynamical System Approach in Partial Differential Equations

本项目的主要目的是研究一些双曲型偏微分方程解的动力学行为，其中包括非线性波动方程、非线性薛定谔方程和可压缩欧拉方程。这些方程被广泛地用来模拟物理、工程、生物、金融等各种问题。PI的目的是为这些方程及其扰动构造特定的解，以理解所建模的现象。该项目考虑的扰动要么具有不同的尺度，要么具有不同的随机性，这需要新的数学思想和工具来处理。成功构造的解不仅将提供新的理论结果，而且将激发高效的数值算法来计算多尺度随机动力系统的解。具体地说，国际和平研究所打算认真研究以下问题。第一类问题是构造三类双曲型偏微分方程解的特殊动力结构，这三类双曲型偏微分方程组包括一组非线性波动方程、三次聚焦薛定谔方程和高维拟线性薛定谔方程。所有这些问题都具有多个时空尺度，这就需要新的数学思想和分析工具来处理。下一个问题是研究Sine-Gordon呼吸子在乘性噪声扰动下的持久性。主要目的是了解随机强迫的影响。需要发展一种新的方法，它源于不变流形、随机动力系统和遍历理论。本项目的最后一个问题是利用不变流形的思想研究等熵和非等熵可压缩流动在某些定态附近解的渐近行为。PI将在几何和分析方面扩展现有的思想和发展技术，不仅为这个问题提供解决方案，而且提供一个新的视角。

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.