RUI: Stability and Interaction of Coherent Structures in Lattice Differential Equations

在空间扩展动力系统的理论中，许多努力都致力于研究活动（或能量或质量）从空间域的一个区域传输到另一个区域的机制，例如行波。 霍夫曼教授将研究格点微分方程中相干结构的稳定性和相互作用，即指数定域行波和脉冲。 这包括（i）离散反应扩散方程中平面前沿的多维稳定性和相互作用;（ii）哈密顿晶格（如Fermi-Pasta-Ulam晶格）中孤立波的碰撞性质;（iii）耗散系统中相干结构的强相互作用，如离散Fitzhugh-Nagumo方程中脉冲的湮灭;（iv）单向格点微分方程（如对流-反应方程的迎风离散化或细胞神经网络模型）中的推进前沿的稳定性。 在格点环境中工作的一个优点是存在性和唯一性是众所周知的，因此人们可以立即转向有关动力学的更详细的问题。 同时，连续时间动力学和离散空间结构之间的相互作用可以产生有趣而微妙的现象，例如面向理性方向的波前与面向非理性方向的波前表现不同。非线性格点微分方程通常太复杂，无法用已知函数写出公式来显式求解。 然而，动力系统的方法可以用来获得不太详细的信息，这往往是足够的科学探究的目的。 例如，考虑以下问题：当晶体生长时，它会接近什么形状？当合金冷却时，其相界将如何演变？给定两个信号沿沿着同一根光纤传播，它们将如何相互作用？一个入侵物种会以多大的速度侵占原生栖息地？主要研究者关注的是发展数学技术来回答这些问题和类似的问题，理想化的物理模型，承认空间离散相干结构。 本科生将参与这项研究。

Much effort in the theory of spatially extended dynamical systems has been devoted to the mechanisms, such as traveling waves, by which activity (or energy or mass) is transported from one region of the spatial domain to another. Professor Hoffman will study the stability and interaction of coherent structures, i.e. exponentially localized traveling waves and pulses, in lattice differential equations (LDE). This includes (i) multidimensional stability and interaction of planar fronts in discrete reaction diffusion equations; (ii) collision properties of solitary waves in Hamiltonian lattices such as the Fermi-Pasta-Ulam lattice; (iii) strong interaction of coherent structures in dissipative systems such as annihilation of pulses in the discrete Fitzhugh-Nagumo equation; (iv) Stability of pushed fronts in unidirectional lattice differential equations such as those obtained from upwind discretization of advection-reaction equations or in cellular neural network models. One advantage of working in the lattice setting is that existence and uniqueness is well-known, hence one can immediately turn to more detailed questions concerning the dynamics. At the same time, the interaction between the continuous temporal dynamics and the discrete spatial structure can give rise to interesting and subtle phenomenon, such as fronts facing rational directions behaving differently from fronts facing irrational directions.Nonlinear lattice differential equations are typically too complex to solve explicitly in the sense of writing down a formula in terms of known functions. However, the methods of dynamical systems can be used to obtain less detailed information and this is often sufficient for the purposes of scientific inquiry. For example, consider the following questions: As a crystal grows, what shape will it approach? As an alloy cools, how will its phase boundaries evolve? Given two signals propagating along the same fiber, how will they interact? At what rate will an invasive species encroach upon native habitat? The Principal Investigator is concerned with developing mathematical techniques to answer these and similar questions for idealized physical models that admit spatially discrete coherent structures. Undergraduate students will participate in this research.